Trojan horse method as surrogate indirect approach to study resonant reactions in nuclear astrophysics
TTrojan horse method as surrogate indirect approach to study resonant reactions innuclear astrophysics
A. M. Mukhamedzhanov, ∗ A.S. Kadyrov, † and D. Y. Pang ‡ Cyclotron Institute, Texas A & M University, College Station, TX 77843, USA Curtin Institute for Computation and Department of Physics and Astronomy,Curtin University, GPO Box U1987, Perth, WA 6845, Australia School of Physics and Beijing Key Laboratory of Advanced Nuclear Materials and Physics,Beihang University, Beijing, 100191, People’s Republic of China
The primary goal of the Trojan horse method (THM) is to analyze resonant rearrangement reac-tions when the density of the resonance levels is low and statistical models cannot be applied. Themain difficulty of the analysis is related with the facts that in the final state the THM reaction in-volves three particles and that the intermediate particle, which is transferred from the Trojan horseparticle to the target nucleus to form a resonance state, is virtual. Another difficulty is associatedwith the Coulomb interaction between the particles, especially, taking into account that the goal ofthe THM is to study resonant rearrangement reactions at very low energies important for nuclearastrophysics. The exact theory of such reactions with three charged particles is very complicatedand is not available. This is why different approximations are used to analyze THM reactions. Inthis review paper we describe a new approach based on a few-body formalism that provides a solidbasis for deriving the THM reaction amplitude taking into account rescattering of the particles inthe initial, intermediate and final states of the THM reaction. Since the THM uses a two-stepreaction in which the first step is the transfer reaction populating a resonance state, we addressthe theory of the transfer reactions. The theory is based on the surface-integral approach and R -matrix formalism. We also discuss application of the THM to resonant reactions populating bothresonances located on the second energy sheet and subthreshold resonances, which are subthresholdbound states located at negative energies close to thresholds. We consider the application of theTHM to determine the astrophysical factors of resonant radiative-capture reactions at energies solow that direct measurements can hardly be performed due to the negligibly small penetrabilityfactor in the entry channel of the reaction. We elucidated the main ideas of the THM and outlinenecessary conditions to perform the THM experiments. CONTENTS
I. Introduction 2II. Transfer reactions populating resonance state inthe surface-integral approach 5A. Theory of transfer reactions populatingresonance states in the surface-integralformalism 5B. Numerical results for reaction O( d , p ) O( / ) 9III. General theory of surrogate reactions in afew-body approach 10A. The amplitude of the breakup reaction in afew-body approach proceeding through aresonance in the intermediate subsystem 11B. Triply DCS 13C. Doubly DCS of transfer reaction populatingresonance state 14D. DWBA amplitude of reaction populatingresonance state 14 ∗ [email protected] † [email protected] ‡ [email protected] E. THM amplitude in plane-wave approach 15IV. Application of DWBA formalism for analysis ofresonant THM reaction of C + C fusion 16A. Kinematics of the THM reaction 17B. DWBA DCS 17C. Renormalization of THM astrophysicalfactors 19D. Astrophysical factors for the C − Cfusion from THM reaction 20E. Concluding remarks 21V. THM for subthreshold resonances 23A. Single-channel single-level case 23B. Two-channel single-level case 24C. S factor for reaction proceeding throughsubthreshold resonance in O 25D. C( α, n ) O reaction 26E. Threshold level 1 / + , l xA = 1 , E x = 6 . C( α, n ) O 28VI. THM for radiative capture reactions 29A. Introduction 29B. Amplitude of indirect resonantradiative-capture reaction 30C. Triply DCS 31 a r X i v : . [ nu c l - t h ] J u l D. Doubly DCS 32VII. Radiative capture C( α, γ ) O via indirectreaction C( Li , d γ ) O 33A. Introduction 33B. Astrophysical factors for C( α, γ ) O 35C. Photon’s angular distributions 36VIII. Summary 38ACKNOWLEDGMENTS 39Appendices 40A. Spectral decomposition of the two-channelGreen’s function 40B. Derivation of the THM radiative captureamplitude 42References 44
I. INTRODUCTION
Often reactions of astrophysical interest are measuredin the laboratory at energies much higher than those rel-evant to stellar processes. After that, an extrapolationdown to stellar energies is performed. This allows one toobtain the astrophysical factors and reaction rates in theso-called Gamow window [1] at which the convolution ofthe Maxwell-Boltzman energy distribution and the crosssection of a reaction has a maximum. However, suchextrapolations may lead to significant uncertainties.There are a few underground laboratories operatingor under construction around the world where the astro-physical reactions can be measured. The most famousis the underground laboratory in Gran Sasso (LUNA),Italy. This facility uses a low-energy accelerator to mea-sure cross sections for reactions involving stable beamsand more than 75 targets at significantly lower energiesthan those previously achieved [2]. But still extrapola-tions to astrophysical energies is usually required. As anexample of the important astrophysical reaction, whichrequires extrapolation of the direct measurements, is the C+ C fusion. There is another problem which plaguesmeasurements of charged particle reaction rates at lowenergies. It is electron screening which increases the crosssection measured in the laboratory compared to the ac-tual nuclear reaction rates in stellar plasma. Similarly,neutron-induced reactions on unstable short-lived nucleialso cannot be presently measured directly in the labo-ratory.Resonant reactions at very low energies are importantpart of the nuclear astrophysical processes occurring instellar environments. The energies at which these reac-tions proceed are much lower than the Coulomb barrier.It makes their cross sections so small that it is very diffi-cult or often impossible to measure them directly in the laboratory. This is owing to the very small barrier pen-etrability factor from the Coulomb + centrifugal forcesproducing an exponential fall off of the cross section asenergy decreases.Indirect techniques, also known as surrogate methods,have been developed over the past several decades toprovide ways to determine reaction rates that cannot bemeasured directly in the laboratory [3]. Important infor-mation that is needed to determine the rates for reactionsthat are dominated by a nuclear resonance is the energyof the resonance and its decay width in the appropriateinitial and final channels. Indirect techniques have beenapplied using both stable and radioactive beams. In thisreview we address the theory and applications of the sur-rogate reactions proceeding through a resonance in thesubsystem. This method allows one to treat a wide vari-ety of astrophysical resonant reactions including resonantrearrangement reactions, reactions proceeding throughsubthreshold resonances and resonant radiative-capturereactions.One of the powerful indirect methods to obtain in-formation about the resonant astrophysical reactions isthe Trojan horse method (THM). The THM is a uniqueindirect technique that allows one to determine the as-trophysical factors and reaction rates for rearrangementand radiative capture resonant reactions by obtainingthe cross section for a binary process through the use ofa surrogate “Trojan horse” (TH) particle, where directmethods are not able to obtain data due to very smallcross sections. The method was originally suggested byBaur [4] but became a powerful and well known indirectmethod under the leadership of Prof. Claudio Spitaleri[3, 5–7]. The THM is used to analyze resonant reactionswhen the level density is low and the statistical theory isnot applicable. Such reactions play an important role innuclear and atomic physics. Here we present a theory ofthe THM based on the few-body formalism.The THM reactions are a subclass of reactions rep-resenting transfer to the continuum followed by rear-rangement reactions x + A → b + B . The THM re-actions leading to 3 particles in the final state are de-scribed by the pole diagram depicted in Fig. 1. Inthis diagram the three-ray vertex corresponds to the vir-tual decay a → s + x and the four-ray vertex describesthe subreaction amplitude x + A → b + B . In theplane-wave approach, the fully differential cross section(DCS), d σ/ dΩ bB dΩ sF d E bB , of the knockout reactionwith three particles in the final state corresponding to thepole diagram is expressed in terms of the doubly DCS,( σ xA → bB / dΩ bB ) HOES , asd σ dΩ bB dΩ sF d E bB = KF | ϕ sx ( p sx ) | (cid:16) d σ xA → bB dΩ bB (cid:17) HOES . (1)All the variables are Galilean invariant. E bB is the b − B relative kinetic energy, p sx is the s − x relative momen-tum, Ω bB is the b − B solid angle and Ω sF is the rel-ative solid angle of s and the center of mass (c.m.) ofthe b + B system. One can also choose a different set ofthe Galilean variables characterizing the fully DCS. KFis a kinematical factor and ϕ sx ( p sx ) is the s − x bound-state wave function in the momentum space. The doublyDCS is half-off-the-energy-shell (HOES) because in theentry channel of the subreaction x + A → b + B particle x transferred from a is off-the-energy-shell (OFES), that is, E x (cid:54) = p x / (2 m x ). In what follows the momentum of thevirtual particle x is denoted by p x to distinguish it fromon-the-energy-shell (OES) momentum k x . Since the par-ticle x is virtual, the relative s − x momentum is denotedas p sx to distinguish it from the OES momentum k sx for which E sx = k sx / (2 µ sx ); µ sx is the reduced mass ofparticles s and x .Let us consider in more details the pole diagram. Theamplitude of this diagram depends on five independentGalilean-invariant variables, but one of them, the rota-tion angle around the momentum of the incident particle A is not significant and can be excluded leaving onlyfour independent invariants in the amplitude of diagram1. Hereafter, we refer to the diagram depicted in Fig. 1as diagram 1. These four invariants determine the com-plete kinematics of the THM reaction A + a → b + B + s .Usually, in THM experiments the angles and momentaof the final state particles b and B are measured. aA sx Bb FIG. 1. The diagram describing the knockout reaction in thePWA.
From the energy-momentum conservation law we get σ x = E x − p x m x = E a − E s − ε sx − ( k a − k s ) m x = E sx − p sx µ sx = − p sx + κ sx µ sx , (2)where σ x is the measure of the deviation of particle x from OES (for the OES particle σ x = 0), E sx = − ε sx , κ sx = √ µ sx ε sx is the a = ( s x ) bound-state wave num-ber, ε sx = m s + m x − m a is the binding energy for thevirtual decay a → s + x and m i is the mass of particle i , µ sx is the reduced mass of particles s and x .Similarly, from the energy and momentum conserva-tion law at the x + A → b + B four-ray vertex we obtain σ x = E xA − p xA µ xA . (3)Then it is evident that, due to the virtual character ofparticle x , E sx (cid:54) = p sx / (2 µ sx ) and E xA (cid:54) = p xA / (2 µ xA ). The real particles are a and A in the entry channel andparticles s, b and B in the exit channel, see Fig. 1. Therelative momenta p sx and p xA are given by p sx = m x k s − m s p x m sx , (4)and p xA = m A p x − m x k A m xA . (5)Here m ij = m i + m j . The relative momentum of real(OES) particles is k ij = m j k i − m i k j m ij . (6)From Eqs. (2) and (3) we get one of the importantenergy-momentum relationships of the THM: E xA = p xA µ xA − p sx µ sx − ε sx . (7)Thus in the THM, due to the virtual character of particle x , we always have p xA / µ xA > E xA . In the quasi-free(QF) kinematics, p sx = 0 and the x − A relative kineticenergy is E xA = p xA (0) µ xA − ε sx . (8)where p xA (0) is the x − A relative momentum in the QFkinematics. It is convenient to rewrite Eq. (7) in thereference frame where k a = 0: E xA = m x m xA E A + k s · k A m xA − k s µ sx − ε sx . (9)We introduce the form factor W a ( p sx ) = (cid:90) d r sx e i p sx · r sx V ( r sx ) ϕ a ( r sx )= − p sx + κ sx µ sx ϕ a ( p sx ) , (10)where ϕ a ( p sx ) is the Fourier transform of the wave func-tion ϕ a ( r sx ) of the bound state a = ( sx ), r sx is theradius-vector connecting the centers of mass of nuclei s and x , V sx ( r sx ) is their interaction potential.If one of the particles s or x is a neutron, the formfactor is regular at p sx + κ sx = 0, that is, ϕ a ( p sx ) = − µ sx W a ( p sx ) p sx + κ sx (11)has a pole at p sx + κ sx = 0. However, if both parti-cles s and x are charged, the potential V sx ( r sx ) includesboth nuclear and Coulomb parts. The latter modifies thebehavior of the form factor at p sx + κ sx = 0 to W a ( p sx ) p sx + κ sx → = [ p sx + κ sx ] η sx ˜ W a ( p sx ) , (12)where ˜ W a ( p sx ) is a function regular at the singular point, η ij = Z i Z j e µ ij κ ij (13)is the Coulomb parameter of the bound state ( i j ), Z i e is the charge of particle i . The system of units inwhich (cid:126) = c = 1 is used throughout the paper. Hencethe Fourier component of the bound-state wave function ϕ a ( r sx ) and, consequently, the amplitude of diagram 1,have the branching point singularity rather than the poleone: ϕ a ( p sx ) p sx +( κ sx ) → = − µ sx ˜ W a ( p sx )[ p sx + κ sx ] − η sx . (14)Therefore, the amplitude of diagram 1 is not a pole inthe presence of the Coulomb interaction at the vertex a → s + x . Only if one of the particles s or x is a neu-tron, that is η sx = 0, the singularity at p sx + κ sx = 0turns into a pole. Strictly speaking, only in this case di-agram 1 can be called a pole diagram. But historically,in the literature diagrams in which a single particle istransferred are called the pole ones. That is why we alsowill use the same name for the amplitude of diagram 1but keeping in mind that for charged particles s and x this amplitude has the branching point singularity ratherthan the pole one.In all the performed THM experiments the orbital an-gular momentum l sx of the bound state ( s x ) is 0. Al-though the singularity at p sx + κ sx = 0 is located inthe unphysical region, the modulus of the amplitude ofdiagram 1 has a peak at p sx = 0, which is called thequasi-free peak. However, the Coulomb s − x interactiondecreases the QF peak, see Eq. (14). Thus the QF kine-matics provides the best condition for the dominance ofthe diagram in Fig. 1.The second THM condition that ensures the valid-ity of the so-called plane-wave impulse approximation(PWIA) follows from Eq. (8): E xA >> ε sx . Then p xA ≈ k xA and the amplitude for the subreaction x + A → b + B reaches its OES limit. Then in Eq. (1) theHOES doubly DCS can be replaced by the OES dou-bly DCS. Note that while the HOES DCS is actually(d σ xA → bB ( k bB , p xA ) / dΩ bB ) HOES , the OES DCS used inthe PWIA is (d σ xA → bB ( k bB , k xA ) / dΩ bB ) OES .Note that in the THM the TH particle a = ( sx ) isusually loosely bound. From the uncertainty principleit follows that for small binding energies the condition p sx (cid:46) κ sx probes distances r sx (cid:38) /κ sx where the nuclearinteraction between s and x is depleted and particle s inthe final state can be treated as a spectator.In practical applications of the THM, experiments areperformed at some fixed value of E aA . To cover a broaderrange of E xA , events that deviate from the QF conditionof p sx = 0 are selected within the interval p sx (cid:46) κ sx .From Eq. (9) it follows that this can be achieved byvarying both direction and magnitude of k s .However, there are serious shortcomings of the PWIA.First, it neglects the Coulomb-nuclear interaction of the particles in the initial and final states of the THM reac-tion, which often becomes very important, especially forheavier particles. The second serious limitation of thePWIA is that the main idea of the THM is to apply itfor astrophysically relevant E xA energies, which may besignificantly smaller than ε sx . At such low E xA the OESPWIA approximation for the doubly DCS (quasi-free ap-proximation) for the subreaction x + A → b + B is notvalid.The indirect TH method is used for obtaining the as-trophysical factors S ( E xA ) for the resonant reactions x + A → F ∗ → b + B. (15)To determine the resonant astrophysical factors for reac-tion (15) a two-step THM resonant a + A → s + F ∗ → s + b + B (16)is used. Here a = ( sx ) is the Trojan horse particle, F ∗ is the intermediate resonance formed by the x + A sys-tem. The THM resonant reactions are described by thediagram depicted in Fig. 2. FIG. 2. The diagram describing the two-step THM resonantreaction in the PWA
In this review we address another approach based onthe surface-integral formalism [8, 9] which does not re-quire the assumption about the quasi-free sub-reaction x + A → b + B and explicitly takes into account the off-shell character of the transferred particle x within thegeneralized R -matrix approach for three-body reactions.We present theoretical foundations of this approach fo-cusing on the THM resonant reactions and provide somecalculations demonstrating our approach for different res-onant reactions.This review is organized as follows. Before address-ing the theory of the two-step THM reactions leading tothree charged particles in the final state, in Section II wepresent the theory for the first step of the THM reac-tion, which is the transfer reaction, based on the surface-integral formalism. The theoretical part is accompaniedby calculations of the O( d, p ) O(1 d / ) transfer reac-tion populating a resonance state where we analyse con-tributions of the internal and external terms in prior- andpost-form adiabatic DWBA (ADWBA). These calcula-tions show why the prior-form ADWBA is more prefer-able than the post one when analysing transfer reactionspopulating resonance states. In Section III we address ageneral theory of the two-step THM resonant reactionsbased on a few-body approach. Section IV presents acritical analysis of the application of the THM methodto obtain the astrophysical factor for carbon-carbon fu-sion, which is of one of the key astrophysical reactions. InSection V we review the theory of the THM for obtainingthe astrophysical factors for the resonant reactions pro-ceeding through subtheshold resonances. In particular,we discuss application of this theory to another key as-trophysical reaction of C( α, n ) O, believed to be theneutron generator in low mass AGB stars. In SectionVI we focus on the application of the THM to measureresonant radiative-capture reactions at energies so lowthat direct measurements can hardly be performed dueto the negligibly small penetrability factor in the entrychannel of the reaction. The developed theory is accom-panied in Section VII by the practical application for theastrophysical reaction C( α, γ ) O via indirect reaction C( Li , d γ ) O . Finally, Section VIII concludes the re-view with a brief summary.
II. TRANSFER REACTIONS POPULATINGRESONANCE STATE IN THESURFACE-INTEGRAL APPROACHA. Theory of transfer reactions populatingresonance states in the surface-integral formalism
Assume that one needs to obtain information about thebinary resonant reaction (15), where F ∗ = ( xA ) is an iso-lated resonance state of the charged particles x and A ,which decays into the channel b + B . The THM can pro-vide this information using the surrogate reaction (16).The first step of the surrogate reaction is the transfer re-action populating the resonance state F ∗ and the second step is decay of the resonance F ∗ → b + B .Since the first step of the reaction (16) is the transferof the particle x populating the resonance state F ∗ , theamplitude of this transfer reaction is divergent. To cal-culate it one needs to know how to handle the divergentamplitude. Two main problems complicate the practicaltheory of stripping into resonance states: (i) the numer-ical problem of the convergence of the matrix element inthe distorted-wave Born approximation (DWBA) whenthe full transition operator is included and (ii) the am-biguity over what spectroscopic information can be ex-tracted from the analysis of transfer reactions populatingthe resonance states. The purpose of this section is to ad-dress these two problems. Our approach is based on thesurface-integral formalism developed in [8, 9]. The reac-tion amplitude is parametrized in terms of the reducedwidth amplitudes related with the resonance widths, in-verse level matrix, boundary condition and channel ra-dius. These are the same parameters that are used inthe conventional R -matrix method [10]. For stripping toresonance states, many-level, and one- and two-channelcases are considered. The theory provides a consistenttool for analyzing binary resonant reactions and strippingto resonance states in terms of the same parameters.Next we apply the surface-integral formalism to de-scribe the transfer reaction populating resonance states.Using the DWBA prior form we obtain the generalizedDWBA R -matrix amplitude for the transfer reaction toresonance states assuming two-channel, multi-level res-onance in the subsystem x + A . The prior-form of theDWBA amplitude for the first step of the THM reaction a + A → s + F ∗ (17)populating the resonance state F ∗ is M DW ( prior ) M F M s ; M A M a ( k sF , k aA ) = (cid:88) m sxA m lxA M x (cid:10) s xA m s xA l xA m l xA (cid:12)(cid:12) J F M F (cid:11)(cid:10) J x M x J A M A (cid:12)(cid:12) s xA m s xA (cid:11) × (cid:10) s sx m s sx l sx m l sx (cid:12)(cid:12) J a M a (cid:11)(cid:10) J s M s J x M x (cid:12)(cid:12) s sx m s sx (cid:11) M DW ( prior ) , (18)with M DW ( prior ) ( k sF , k aA ) = (cid:10) χ ( − ) sF Υ ( − ) xA | ∆ V aA | ϕ a χ (+) aA (cid:11) , (19)where χ (+) aA is the distorted wave describing the relativemotion in the initial channel with the outgoing bound-ary condition, χ ( − ) sF is the distorted wave describing therelative motion in the final channel with the incomingboundary condition, k ij is the i − j relative momentum.Also, s ij ( m s ij ) is the channel spin (its projection) in thechannel i − j , l ij ( m l ij ) is the relative orbital angularmomentum of particles i and j , J i ( M i ) is the spin (itsprojection) of particle i . The transition operator in the prior form is∆ V aA = U sA + V xA − U aA . (20)Here, U ij is the i − j optical potential and V xA = (cid:10) ϕ A | V xA | ϕ A (cid:11) is the mean-field real potential support-ing the resonance state in the system F = ( x A ), ϕ i is the bound-state wave function of nucleus i . Υ ( − ) xA = (cid:10) ϕ A | Ψ ( − ) xA (cid:11) is the overlap of the resonant wave functionΨ ( − ) xA in the channel x + A and the bound-state wave func-tion of nucleus ϕ A . We neglect the internal degrees offreedom of nucleus x .One can also use the post-form of the DWBA ampli-tude M DW ( post ) ( k sF , k aA ) = (cid:10) χ ( − ) sF Υ ( − ) xA | ∆ V sF | ϕ a χ (+) aA (cid:11) , (21)with the post-form transition operator given as∆ V sF = U sA + V sx − U sF , (22)where V sx is the s − x potential supporting the a = ( sx )bound state. However, we prefer to use the prior-formdue to the faster convergence over variable r xA , which isthe coordinate vector conjugate of the relative momen-tum k xA (see section II B).The matrix element in Eq. (19) involves integrationover variable r xA . Following Refs. [9] and [11] we splitthe matrix element (19) into an internal part with r xA ≤ R ch and the external part with r xA > R ch : M DW ( prior ) ( k sF , k aA ) = M DW ( prior ) int ( k sF , k aA )+ M DW ( prior ) ext ( k sF , k aA ) , (23)with M DW ( prior ) int ( k sF , k aA )= (cid:10) χ ( − ) sF Υ ( − ) xA | ∆ V aA | ϕ a χ (+) aA (cid:11)(cid:12)(cid:12)(cid:12) r xA ≤ R ch (24)and M DW ( prior ) ext ( k sF , k aA )= (cid:10) χ ( − ) sF Υ ( − ) xA | ∆ V aA | ϕ a χ (+) aA (cid:11)(cid:12)(cid:12)(cid:12) r xA >R ch . (25)Here R ch is the channel radius, which is chosen so that at r xA > R ch the nuclear x − A interaction can be neglectedand the V xA potential can be replaced by the Coulombterm V CxA . Since s and x are close to each other due tothe presence of the bound-state wave function ϕ sx the V sA potential can also be approximated by the Coulombterm. Then in the external region over the variable r xA we have M DW ( prior ) ext = (cid:10) Ψ C ( − ) k sF Υ ( − ) xA (cid:12)(cid:12) V CsA + V CxA − U CaA (cid:12)(cid:12) ϕ sx Ψ C (+) k aA (cid:11)(cid:12)(cid:12)(cid:12) r xA >R ch . (26)In this amplitude Υ ( − ) xA can be replaced by the resonantwave function in the external region given by Eq. (36)from [12]:˜ φ R ( xA ) ( r xA ) = e − i δ psxAlxAJF ( k xA ) ) (cid:114) µ xA k xA ) Γ ( xA ) × O ∗ ( k xA ) , r xA ) r xA , (27) where δ ps xA l xA J F ( k xA ) ) is the x − A potential (non-resonant) scattering phase shift with the quantum num-bers of the resonance F ∗ , k xA ) is the real part of the x − A resonance momentum, O ( k xA ) , r xA ) is the outgo-ing Coulomb scattering wave.The external matrix element M DW ( prior ) ext in the priorform is small and in some cases with a reasonable choiceof the channel radius R ch ( xA ) can even be neglected [9].However, it is important for analysis of the stripping toresonance states because the external part in the postform does not converge. In this sense the usage of theprior form in the external part has a clear benefit.The splitting of the amplitude into the internal andexternal parts in the subspace over the x − A coordi-nate r xA is necessary to rewrite the prior DWBA am-plitude in the generalized R -matrix approach for strip-ping to resonance states. The main contribution to theprior form amplitude M DW ( prior ) comes from the inter-nal part M DW ( prior ) int . Since the internal part is given bya volume integral, its calculation requires the knowledgeof Υ ( − ) xA in the internal region. The model dependence ofthis function in the nuclear interior ( r xA ≤ R ch ), wheredifferent coupled channels contribute, brings one of themain difficulties and leads to the main uncertainty inthe calculation of the internal matrix element. Using asurface-integral representation we can rewrite the volumeintegral of the internal matrix element in terms of the vol-ume integral in the post form and the dominant surfaceintegral taken over the sphere at r xA = R ch . With a rea-sonable choice of the channel radius R ch the contributionfrom the internal volume integral in the post form canbe minimized to make it significantly smaller than thesurface matrix element. The latter can be expressed interms of the R -matrix parameters such as the observablereduced width amplitude (observable resonance width),boundary condition and channel radius.For the purpose of transforming M DW ( prior ) int into asurface integral in the subspace over variable r xA werewrite the transition operator in the internal region as∆ V aA = U sA + V xA − U aA = [ V xA + U sF ] + ( U sA + V sx − U sF ) − [ V sx + U aA ] . (28)The bracketed transition operators are the potential op-erators in the Schr¨odinger equations for the initial andfinal channel wave functions. Hence, for the internal priorform of the DWBA we obtain M DW ( prior ) int ( k sF , k aA ) = M DW ( post ) int ( k sF , k aA )+ M DWS ( k sF , k aA ) , (29)where M DW ( post ) int ( k sF , k aA )= (cid:10) χ ( − ) sF Υ ( − ) xA | ∆ V sF | ϕ a χ (+) aA (cid:11)(cid:12)(cid:12)(cid:12) r xA ≤ R ch (30)is the internal post-form DWBA amplitude and M DWS ( k sF , k aA ) = − (cid:10) χ ( − ) sF Υ ( − ) xA | ←− ˆ T − −→ ˆ T | ϕ a χ (+) aA (cid:11) = − (cid:10) χ ( − ) sF Υ ( − ) xA | ←− ˆ T xA − −→ ˆ T xA | ϕ a χ (+) aA (cid:11) (31)is the surface term. Here ˆ T is the total kinetic energyoperator in the c.m. of the reaction and ˆ T xA is the kineticenergy operator of the relative motion of particles x and A . Arrows indicate the direction in which the differentialkinetics energy operator acts. It is important to notethat with a proper choice of the optical potential U sF the matrix element M DW ( post ) int can be minimized so thatits model dependence would not have impact on the totalmatrix element M DW ( prior ) . That is why in what followswe disregard M DW ( post ) int .We show now how to simplify M DWS ( k sF , k aA ) by reducing it to a surface-integral form. In the surface-integral form the resonance overlap function Υ ( − ) xA is re-placed with the ( xA ) real resonance wave function ˜ φ R ( xA ) given by Eq. (27).We use a three-body model of the constituents s, x and A , all assumed to be structureless particles. In amore general approach we need to introduce the projec-tion operators to ensure that particles x and A are in theground states in the intermediate states of the transferreaction. In this case the bound-state wave function ϕ a and the resonance wave function ˜ φ R ( xA ) should be re-placed by the overlap functions. These overlap functionscan be approximated by the product of the two-bodywave functions and the square roots of the correspond-ing spectroscopic factors. Since in the THM only theenergy dependence of the DCS are measured, these spec-troscopic factors can be dropped. Here we also use thetwo-body wave functions rather than the overlap func-tions. Then M DW ( prior ) S = (cid:10) Ψ C ( − ) k sF ˜ φ R ( xA ) (cid:12)(cid:12) ←− ˆ T xA − −→ ˆ T xA (cid:12)(cid:12) ϕ a Ψ C (+) k aA (cid:11)(cid:12)(cid:12)(cid:12) r xA = R ch = R ch µ xA (cid:90) d r sF Ψ C (+) − k sF ( r sF ) (cid:90) d Ω r xA × (cid:104) ϕ a ( r sx ) Ψ C (+) k aA ( r aA ) ∂φ R ( xA ) ( r xA ) ∂r xA − φ R ( xA ) ( r xA ) ∂ϕ a ( r sx )Ψ C (+) k aA ( r aA ) ∂r xA (cid:105)(cid:12)(cid:12)(cid:12) r xA = R ch . (32)Then the external part of the amplitude can be expressed in terms of the resonance width Γ xA and reduces to M DW ( prior ) ext = e i δ psxAlxAJF ( k xA ) ) (cid:114) µ xA k xA ) Γ ( xA ) (cid:10) Ψ C ( − ) k sF O ∗ ( k xA ) , r xA ) r xA (cid:12)(cid:12) V CsA + V CxA − U CaA (cid:12)(cid:12) ϕ a Ψ C (+) k aA (cid:11)(cid:12)(cid:12)(cid:12) r xA >R ch . (33)Thus, the internal matrix element consists of two terms, the internal post-form Coulomb DWBA amplitude M DW ( post ) int and the surface term M DW ( prior ) S . The internal Coulomb or Coulomb+nuclear DWBA in the post formshould be small due to the highly oscillatory behavior of the binned resonance wave functions (this will be demon-strated in part IV B). Note also that the smaller the resonance energy, the smaller is the contribution of the internalregion. Then the dominant contribution to the matrix element M DW ( prior ) comes from the surface term M DW ( prior ) S and M DW ( prior ) ext .We transform now the surface matrix element into zero-range DWBA amplitude. To this end we use r aA = r xA + m s m sx r sx , r sF = m A m xA r xA + r sx . (34)Rewriting the wave functions Ψ C (+) k aA ( r aA ) and Ψ C (+) − k sF ( r sF ) in the momentum space we get M DW ( prior ) S = R ch µ xA (cid:90) d r sF (cid:90) d p sF (2 π ) (cid:90) d p aA (2 π ) Ψ C (+) − k sF ( p sF ) Ψ C (+) k aA ( p aA ) ϕ a ( r sx ) e − i p sx · r sx × (cid:90) d Ω r xA (cid:104) e i p xA · r xA ∂φ R ( xA ) ( r xA ) ∂r xA − φ R ( xA ) ( r xA ) ∂ e i p xA · r xA ∂r xA (cid:105)(cid:12)(cid:12)(cid:12) r xA = R ch , (35)where p xA = p aA − m A m F p sF , p sx = p sF − m s m a p aA . (36)Taking into account the fact that r xA = R ch is larger than the nuclear interaction radius we replace the relativemomentum p xA with the momentum k xA = k aA − m A m F k sF , which is expressed in terms of the OES momenta k aA and k sF . However, the x − A relative momentum k xA is OFES because the transferred particle x is OFES.We consider M DW ( prior ) S at the real part of the ( xA ) resonance energy, i.e., k xA = k xA ) and k sF = k sF ) . Thenreturning to the coordinate-space representation for M DW ( prior ) S we get M DW ( prior ) S = R ch µ xA M DW ZR ( prior ) (cid:90) d Ω r xA (cid:104) e − i k xA · r xA ∂φ R ( xA ) ( r xA ) ∂r xA − φ R ( xA ) ( r xA ) e − i k xA · r xA ∂r xA (cid:105)(cid:12)(cid:12)(cid:12) r xA = R ch . (37)Here, M DW ZR ( prior ) = (cid:90) d r sx Ψ C (+) − k sF ) ( r sx ) ϕ a ( r sx ) × Ψ C (+) k aA (cid:18) m s m a r sx (cid:19) (38)is the DWBA amplitude, which does not depend on theresonant wave function ˜ φ R ( xA ) and V xA potential. Thisequation looks like the zero-range DWBA (ZRDWBA). However, in contrast to the standard zero-range approx-imation, Eq. (38) can be used for arbitrary value of theorbital momentum of the resonance state ( xA ). Notethat replacing in Eq. (38) the distorted waves by theplane waves leads to the PWA introduced in [3] and usedin [13].Integrating in Eq. (37) over Ω r xA and using again Eq.(27) for the external resonant wave function we arrive atthe surface term of the DWBA reaction amplitude singledout using the surface-integral formalism: M DW ( prior ) S = i − l xA e − i δ psxAlxAJF ( k xA ) ) (cid:115) µ xA k xA ) Γ xA O l xA ( k xA ) R ch ) j l xA ( k xA ) R ch ) × W l xA Y l xA ,m lxA ( (cid:98) k xA ) ) M DW ZR ( prior ) . (39)Note that from the energy conservation in the transferreaction E aA − ε sx = E sF + E xA (40)it follows that when E xA approaches the complex reso-nance energy E R ( xA ) while the energy E sF reaches thecomplex energy E R ( sF ) , which corresponds to the reso-nance in subsystem F = ( xA ). Hence k sF ) is the realpart of the complex s − F relative momentum k R ( sF ) corresponding to the k xA ) , which is the real part of thecomplex resonance x − A momentum k R ( xA ) . The off- shell factor W l xA is written as W l xA = (cid:104) j l xA ( k xA ) r xA ) (cid:104) R ch ∂ ln[ O l xA ( k xA ) r xA )] ∂r xA − (cid:105) − R ch ∂ ln [ j l xA ( k xA ) r xA )] ∂r xA (cid:105)(cid:12)(cid:12)(cid:12) r xA = R ch , (41)where O l xA is the outgoing spherical wave and j l xA is thespherical Bessel function.Equation (39) is a very important result. It containsthe off-shell factor W l xA reflecting the virtual character ofthe transferred particle x . It also contains the boundaryconditions expressed in terms of the logarithmic deriva-tives and generalizes the R matrix for method for binaryreactions. Hence in the surface-integral formalism theOSE limit of the transferred particle is not required.Thus in the surface-integral formalism the dominantcontribution to the DWBA amplitude of reaction (17)in the prior form is given by the sum of the dominantsurface and external terms and the final expression forthe prior-form DWBA amplitude is M ( prior ) M F M s ; M A M a ( k sF ) ˆk sF , k aA ) ≈ (cid:88) m sxA m lxA M x (cid:10) s xA m s xA l xA m l xA (cid:12)(cid:12) J F M F (cid:11)(cid:10) J x M x J A M A (cid:12)(cid:12) s xA m s xA (cid:11) × (cid:10) s sx m s sx l sx m l sx (cid:12)(cid:12) J a M a (cid:11)(cid:10) J s M s J x M x (cid:12)(cid:12) s sx m s sx (cid:11) (cid:104) M DW ( prior ) S + M DW ( prior ) ext (cid:105) . (42)This expression can be used for the analysis of the trans-fer reaction (17). To analyze the THM reaction (16) onecan use Eq. (42) with the added part describing the sec-ond step of the THM reaction.An important feature of Eq. (42) is that it isexpressed in terms of the resonance width in thedecay channel F ∗ → x + A . The amplitude M ( prior ) M F M s ; M A M a ( k sF ) ˆk sF , k aA ) is the amplitude of thetransfer reaction (17) in the surface-integral approach,which is the first step of the THM reaction. If we addthe second step then we will be able to express the THMreaction amplitude in terms of the OSE astrophysical fac-tor despite the presence of the off-shell factor. There isone more important feature of Eq. (42) to be noted. Theamplitude M ( prior ) M F M s ; M A M a ( k sF ) ˆk sF , k aA ), according toEqs. (39) and (33), is expressed in terms of √ Γ xA . Aswe will see below in Section III C, this allows one to sin-gle out the astrophysical S factor from the THM doublyDCS. B. Numerical results for reaction O( d , p ) O( / ) In this section we present DWBA DCS calculations fora particular transfer reaction populating the resonancestate, which corroborate our theoretical findings al-though the code for the surface-integral formalism is notyet available. We select the reaction O( d, p ) O(1 d / ) at E d = 36 MeV populating a resonance state of energy E x = 5 .
085 MeV, which corresponds to the resonancelevel at 0 .
94 MeV. In all the calculations shown below weuse the single-particle approach for the n − A resonantscattering wave function calculated in the Woods-Saxonpotential with the radial parameter r = 1 .
25 fm anddiffuseness a = 0 .
65 fm.We compare the post- and prior-form calculations fol-lowing the procedure developed in Section II. The postand prior adiabatic distorted-wave approach (ADWA)and prior coupled-channel Born approximation (CCBA)are used for comparison. For the prior ADWA ampli-tude we use Eq. (19). The prior ADWA is the standardprior DWBA in which the initial deuteron potential isgiven by the sum of the optical U pA and U nA potentialscalculated at half of the deuteron energy using the zero-range Johnson-Sopper prescription [21]. The resonancescattering wave function Ψ ( − ) nA is taken in the form of thebinned function for the resonant partial wave [22] whichhas asymptotically both incident and outgoing waves.Hence, the resonant wave function used in FRESCO codeis different from the one considered in Eq. (137). Thelatter has the outgoing wave and used in subsection III D.In the CCBA the final n + A resonant wave func-tion is given by the binned resonant wave function χ ( res )( − ) k pF ( k nA ) (r pF ) which is coupled with two bound statesin O: the ground state 1 d / and the first excited state2 s / . Schematically we can write the final-state wavefunction in CCBA asΨ CDCC ( − ) f (r pF , r nA ) = ϕ (0) nA ( r nA ) χ (0)( − ) k pF (r pF ) + ϕ (1) nA ( r nA ) χ (1)( − ) k pF (r pF ) + ψ ( res )( − ) k nA , l nA =3 ( r nA ) χ ( res )( − ) k pF ( k nA ) (r pF ) . (43)Here, for simplicity, we omitted spins. The radialand momentum spherical harmonics are absorbed into ψ ( res )( − ) k nA ( r nA ).In Fig. 3 we present the ratio R x of the ADWA andCCBA DCSs for the deuteron stripping to resonance O( d, p ) O(1 d / ) at E d = 36 MeV calculated at zeroproton scattering angle with cutoff over r nA to the totalADWA and CCBA DCSs calculated at zero proton scat-tering angle without cut-off. This figure provides a veryimportant justification for the surface integral formalism used in subsection II A. The dark blue short dashed lineand light-blue long dashed-dotted line show the ratios R x of the zero-angle prior ADWA and CCBA DCSs, re-spectively, to the full DCS, in which the radial integralover r nA is calculated for r nA ≥ r minnA while r maxnA is ex-tended to infinity. Note that r minnA ( r maxnA ) determines thelow-limit (upper limit) of the matrix element radial inte-gral. As we see, when r minnA varies from 0 until 2 fm theprior DCS does not change. Hence we can start the inte-gration in the radial matrix element from r minnA = 2 fm.0The fact that for r minnA ≥ r maxnA = 5 fm and we canuse the radial integral of the matrix element in the in-terval 2 ≤ r nA ≤ r minnA = 0 to r maxnA , where r maxnA varies. From Fig. 3 it follows thatwe can use r maxnA = 5 fm. We also see that at r maxnA = 2fm the prior DCS vanishes. Again, as concluded for thedark-blue short-dashed and light-blue long dashed-dottedlines, we can calculate for the magenta dotted and greendashed lines that the radial matrix element can be calcu-lated from r minnA = 2 fm to r maxnA = 5 fm. The surface term(39) obtained in subsection II A allows one to replace thevolume matrix element calculated from r minnA = 2 fm to r maxnA = 5 fm by the surface term calculated at R ch = 5fm. The surface term approximation is based on the as-sumption that the internal post form at r nA ≤ r nA ≥ III. GENERAL THEORY OF SURROGATEREACTIONS IN A FEW-BODY APPROACH
In the previous section we developed the DWBA the-ory of the transfer reactions populating a resonance state.In this section within a genuine three-body approach weconsider theory of transfer reactions proceeding througha resonance state in the intermediate subsystem and lead-ing to three particles in the final state [12]. All the initial,intermediate and three-body final-state interactions aretaken into account. We derive the fully differential crosssection (DCS) in terms of the transfer reaction amplitudeobtained in the previous section.We begin by considering the surrogate reaction (16).In nuclear physics, such surrogate reactions are used inthe THM allowing one to obtain a vital astrophysicalinformation about a binary resonant subreaction [3, 4,6, 14]. In the THM the initial channel is a + A , where a = ( sx ) is the Trojan Horse (TH) particle, rather thanjust x + A . Collision a + A is followed by the transferreaction a + A → s + F ∗ populating a resonance state F ∗ in the subsystem F = x + A , which decays to anotherchannel b + B . In the THM reaction (16), this results inthree particles b + B + s in the final state, rather thanthe two-particle final state b + B in the binary reaction.The indirect THM allows one to bypass the Coulomb R X r nA (fm) FIG. 3. (Color online) Dependence of the normalized ADWAand CCBA DCS ratios R x on r nA for the deuteron strippingto resonance O( d, p ) O(1 d / ) at E d = 36 MeV. Darkblue short dashed line and light-blue long dashed-dotted line- the ratios R x of the zero-angle prior ADWA and CCBADCSs, respectively, in which the radial integral over r nA iscalculated for r nA ≥ r minnA , to the full DCS. Similarly, ma-genta dotted and green dashed lines are the ratios R x of thezero-angle prior ADWA and CCBA DCSs, respectively, inwhich the radial integral over r nA is calculated in the interval0 ≤ r nA ≤ r maxnA , to the full DCS. The red solid line is the R x dependence on r maxnA calculated for the post ADWA form.Hence r nA on the abscissa is r minnA for the blue short and longdashed lines and r maxnA for the dotted magenta, dashed greenand solid red lines. First published in [11]. barrier issue in the system x + A by using reaction (16)rather than the binary reaction (15). However, the em-ploying of the surrogate reaction leads to complicationsof both experimental and theoretical nature. Experimen-tally, coincidence experiments are needed. From the the-oretical point of view, the presence of the third particle s can affect the binary reaction (15), especially if theparticle s is charged. In this case, it interacts with theintermediate resonance F ∗ and with the final products b and B via the Coulomb forces, which are very importantat low energies and larger charges of the participatingnuclei.The THM reactions induced by collision of the Nand C nuclei lead to three charged particles in the finalstate. Recently, the THM reaction N + C → d + Mg ∗ → d + b + B, (44)where b = p and α , B = Na and Ne, respectively, wasused to obtain an information about the C + C fusion[13], which is a very important reaction in nuclear astro-physics [1]. In this reaction the Coulomb effects shouldhave dramatic effect on the cross section. In particular,the post-collision Coulomb interaction may have a crucialimpact on the DCSs [15].We use a few-body approach to derive the amplitude1and the fully DCS of the surrogate reaction (16) takinginto account the Coulomb interaction of the particle s with the resonance F ∗ in the intermediate state and theCoulomb interactions of s with b and B in the final state.Our final results reveal a universal effect of the Coulombinteraction important for both nuclear and atomic surro-gate reactions. For nuclear surrogate reactions the short-range nuclear interactions should be taken into accountalongside the long-range Coulomb interactions. However,the consideration of the nuclear surrogate reaction is sim-plified due to the fact that the nuclear rescattering effectsin the intermediate s − F ∗ state and within the s − b and s − B pairs in the final state give rise to diagrams that canbe treated as a background and disregarded for narrowresonances.The main goal of the THM is to investigate an energybehavior of the fully DCS which is needed to determinethe astrophysical factors [3]. The obtained fully DCSallows one to investigate the Coulomb effects on the res-onance line shape both in atomic and nuclear collisions.To calculate the energy dependence of the THM fullyDCS one needs to calculate the DCS of the transfer reac-tion. To treat the final-state three-body Coulomb effectswe use the paper [16]. In this work the formalism of thethree-body Coulomb asymptotic states (CAS) was usedto calculate the reaction amplitudes with three chargedparticles in the final state. A. The amplitude of the breakup reaction in afew-body approach proceeding through a resonancein the intermediate subsystem
Let us consider the surrogate reaction (16), which is thetwo-step THM reaction. The difficulty with the analysisof such a reaction stems from the fact that the resonancedecays into the channel b + B , which is different from theentry channel x + A of the resonant subreaction. As wementioned earlier, the goal of the THM is to extract fromthe TH reaction the astrophysical factor for the resonantrearrangement reaction (15).To simplify considerations we neglect the spins of theparticles with the relative orbital angular momenta l inthe pairs x + A and b + B set l = 0. For brevity of thenotation we omit the angular momenta keeping in mindthat their values are zero. The starting expression for thebreakup reaction amplitude in the c.m. of the few-bodysystem can be written as M = (cid:10) ψ (0) k B , k b (cid:12)(cid:12)(cid:10) X f (cid:12)(cid:12) U A (cid:12)(cid:12) ϕ a ϕ A (cid:11)(cid:12)(cid:12) ψ (0) k aA (cid:11) , (45)where X f = ϕ B ϕ b . The particle s in the THM is aspectator and we can disregard its internal structure andtreat it as a structureless point-like particle. Wave func-tion ψ (0) k aA represents the plane-wave describing the rela-tive motion of the noninteracting particles a and A in theinitial state of the THM reaction, ψ (0) k B , k b is the three-body plane wave of particles s, b and B in the final state, k i is the momentum of particle i . Note that for the mo-ment we use for the charged particles screened Coulombpotentials. The transition operator U A corresponds tothe breakup reaction from the initial channel a + A tothe final three-body channel s + b + B . The two-fragmentpartition α + ( β γ ) with free particle α and the boundstate ( β γ ) is denoted by the free particle index α .The transition operator U A satisfies the equation U A = V f + V f G V sx . (46)Here and in what follows we use the following notations: V i j = V Ni j + V Ci j is the interaction potential between parti-cles i and j given by the sum of the nuclear and Coulombpotentials, V = V + V , and V f = V bB + V sB + V sb .As we mentioned above, the difficulty of the problemis due to the fact that in the TH process the final three-body system s + b + B , which is formed after the resonancedecay F ∗ → b + B , is different from the initial three-bodysystem s + x + A before the resonance F ∗ was formed.This change is caused by the rearrangement resonant re-action (15). The full Green’s function resolvent in Eq.(46) is G = 1 z − ˆ T sF − V sF − ˆ H F , (47)where V sF is the s − F interaction potential, ˆ H F is theinternal Hamiltonian of the system F = x + A = b + B .Our final goal is to single out the resonance term in thesubsystem F = x + A = b + B , which generates a peak inthe fully DCS of the breakup reaction (16), to obtain thereaction amplitude corresponding to the diagrams shownin Fig. 4.First we single out the resonance term with all theCoulomb rescatterings in the initial, intermediate andfinal states. To this end one can rewrite U A as U A = V f + V f G V sx = V f + ( V bB + V CbB ) G V sx + V NbB
G V sx , (48)where V N,CbB = V N,Csb + V N,CsB . The resonance term inthe subsystem F , which can be singled out from the fullGreen’s function G, will be smeared out by nuclear in-teractions V NbB = V Nsb + V NsB . Hence the term V NbB
G V sx does not produce the resonance peak in the TH reactionamplitude generated by the resonance in the subsystem F and can be treated as a background. That is why theonly term in Eq. (48), which is responsible for a reso-nance behavior of the TH reaction amplitude caused bythe resonance in the subsystem F, is U R0 A = ( V bB + V CbB ) G V sx . (49)Note that the Coulomb potentials V CsB and V Csb do notsmear out the resonance in the subsystem F, which canbe singled out from G [17].2 (b) a BFFA bsx sa = (a) sa FFAA F s s B bsa A X FIG. 4. The diagrams describing the TH mechanism includingthe Coulomb interactions in the initial, final and intermedi-ate states. The grey bulb on the left side of diagram (a) isthe Coulomb a + A scattering in the initial channel describedby the Coulomb scattering wave function. The grey rectan-gle is the Green’s function in the final state describing thepropagation of the system s + F , where F is the resonance.The grey bulb on the right side describes the intermediatestate three-body Coulomb interaction given by the three-bodyCoulomb wave function. In diagram (b), the Green’s functionis replaced by its spectral decomposition, which includes theCoulomb scattering wave functions (grey bulbs) describingthe Coulomb rescattering of the spectator s in the intermedi-ate state. Substituting U R0 A into the matrix element (45) insteadof U A leads to M (cid:48) = (cid:10) Φ C ( − ) bB (cid:12)(cid:12)(cid:10) X f (cid:12)(cid:12) V bB G V sx (cid:12)(cid:12) ϕ a ϕ A (cid:11)(cid:12)(cid:12) ψ (0) k aA (cid:11) , (50)where V sx = V CsB + V Csb . To obtain Eq. (50) the two-potential formula is applied. The wave function Φ C ( − ) bB in the bra state is a solution of the Schr¨odinger equationΦ C ( − ) ∗ bB (cid:0) E f − ←− ˆ T sbB − V CsB − V Csb (cid:1) = 0 , (51)with the incoming-wave boundary condition. Here theoperator ←− ˆ T sbB acting to the left is the total kinetic en-ergy operator of the three-body system s + b + B and E f is the total kinetic energy of the system s + b + B in thec.m. system.Now rewriting the potential V sx as V sx = V Nsx + V Csx and applying again the two-potential formula, we reduceEq. (50) to M (cid:48) = (cid:10) Φ C ( − ) bB (cid:12)(cid:12)(cid:10) X f (cid:12)(cid:12) V bB G V
Nsx (cid:12)(cid:12) ϕ a ϕ A (cid:11)(cid:12)(cid:12) Ψ C (+) k aA (cid:11) . (52)Here, Ψ C (+) k aA is the a + A Coulomb scattering wave func-tion, in the Coulomb potential V Csx = V CsA + V CxA with theoutgoing-wave boundary condition.It is important to explain again why we are takinginto account only the Coulomb distortions in the ini-tial and final states rather than the Coulomb plus nu-clear distortions. Firstly, as it will be shown below theCoulomb rescatterings transform the resonance pole into the branching point without smearing out the resonancepeak. Secondly, in the THM, as used currently [3], onlythe energy dependence of the extracted astrophysical fac-tor is measured while its absolute value is determined bythe normalization to the available direct data. Thirdly,the Coulomb rescatterings in the initial, intermediateand final states, in contrast to the nuclear distortions,can significantly modify the energy dependence of theTHM DCS for the reactions at sub-Coulomb and near theCoulomb barrier energies and, therefore, must be takeninto account.In the first step of the THM reaction the particle x ,being in the ground state, is transferred from the boundstate a = ( s x ) to the nucleus A forming a resonance F ∗ = x + A . The THM reaction amplitude, in whichthe particle x is transferred in the ground state, can beobtained from Eq. (52): M (cid:48) = (cid:10) Φ C ( − ) bB (cid:12)(cid:12)(cid:10) X f (cid:12)(cid:12) V bB G V
Nsx (cid:12)(cid:12) X i (cid:11)(cid:12)(cid:12) I ax Ψ C (+) k aA (cid:11) , (53)where X i = ϕ x ϕ A and ϕ x is the bound-state wave func-tion in the ground state. We also introduce the overlapfunction of the bound-state wave functions of nuclei a and x : I ax ( r sx ) = (cid:10) ϕ x ( ξ x ) (cid:12)(cid:12) ϕ a ( ξ x ; r sx ) (cid:11) , (54)which is the projection of the bound-state wave function ϕ a on the two-body channel s + x . The integration in thematrix element is carried over the internal coordinates ξ x of nucleus x . Since we assume that the relative s − x orbital angular momentum in the bound state a = ( sx )equals zero the overlap function depends on r sx , the ra-dius connecting the c.m. of nuclei s and x , rather thanon r sx . We recall that the internal degrees of freedom ofthe spectator s are neglected.Singling out the resonance contribution F ∗ in the in-termediate state of the two-step THM reaction (16) re-quires lengthy tedious transformations. These are givenin Appendix A.As described in Appendix A, the amplitude of two-stepTHM reaction Eq. (16) reads as M R = − i π (cid:114) µ bB k R( bB ) e i δ psxAlxAJF ( k bB ) ) Γ / bB M ( tr ) E bB ) − E bB − i Γ2 × N C ( E bB , ζ ) . (55)Here M ( tr ) = M ( prior ) M F M s ; M A M a ( k sF ) ˆk sF , k aA ) is the am-plitude of the reaction (17) , which is the first step ofthe THM reaction (16). M ( prior ) M F M s ; M A M a ( k sF ) ˆk sF , k aA )is given by Eqs. (42), (37) and (33).3Also N C ( E bB , ζ ) = Γ(1 + i η bs ) Γ(1 + i η Bs )Γ (cid:16) i [ η bs + η Bs ] (cid:17) × F (cid:16) − iη Bs , − iη bs , − (cid:17) ( − γ (0)) i η bs × ( − ν ) i η Bs (cid:18) E bB ) − E bB − i Γ2 (cid:19) − i ζ (56)is the Coulomb renormalization factor which is equal tounity when the Coulomb interactions are turned off. Notethat Eq. (55) is the three-body generalization of theresonant DWBA reaction amplitude obtained in [9]. Itexplicitly takes into account the Coulomb s − F interac-tion in the intermediate state and the final-state s − b − B three-body Coulomb interaction not accounted for in Ref.[9]. Also here ζ = η bs + η sB − η R , (57) ν = − k sF ) ˆk sF · k sB + k sF ) k sB , γ (0) = − k sF ) ˆk sF · k sb + k sF ) k sb can be obtained from Eqs. (62)-(64)from [12] assuming the narrow resonance. In addition, η R = Z s Z F α µ sF /k R is the Coulomb parameter of theinteraction of the spectator s and the resonance F ∗ inthe intermediate state, α = e / ( (cid:126) c ) is the fine-structureconstant, η ij = ( Z i Z j α µ ij /k ij is the Coulomb parame-ter of the particles i and j , E sF − E R = E R( bB ) − E bB = E bB ) − E bB − i Γ / E R = k / (2 µ sF ), k R is given byEq. (142), E sF is relative kinetic energy of the particle s and the c.m. of the system b + B in the final state, and E bB is the b − B relative kinetic energy in the final state.Thus, using a few-body approach we derived the ex-pression for the TH reaction proceeding though the reso-nance in the binary subsystem. The intermediate s − F ∗ and the final-state three-body Coulomb interactions havebeen taken into account explicitly using the three-body formalism. The THM reaction amplitude proceedingthrough the intermediate resonance in the binary sub-system F has in the denominator the resonant energyfactor E bB ) − E bB − i Γ /
2. However, a conventionalBreit-Wigner resonance pole ( E bB ) − E bB − i Γ / − isconverted into the branching point singularity ( E bB ) − E bB − i Γ / − − i ζ . This transformation of the resonancebehavior of the THM reaction amplitude is caused by theCoulomb interaction of the particle s with the resonancein the intermediate state and with products b and B ofthe resonance in the final state.The final-state Coulomb effects have a universal fea-ture and should be taken into account whether one con-siders nuclear reactions leading to the three-body finalstates or in atomic processes. If η bs , η Bs and Re( η R )have the same sign, the Coulomb s − F ∗ interaction inthe intermediate state weakens the impact of the finalstate Coulomb s − b and s − B interactions. This isbecause the intermediate state Coulomb parameter η R is subtracted from the final-state Coulomb parameters η bs + η Bs , see Eq. (57). For example, if the Coulomb s − F ∗ interaction in the intermediate state is turned off,that is η R = 0, the resonance behavior of the TH reactionamplitude coincides with that from Ref. [18] where theangular and energy dependences of the electrons formedfrom the autoionization resonances induced by collisionsof fast protons with atoms were investigated. B. Triply DCS
Now we present some useful equations for the fully anddoubly DCSs which can be used in the analysis of theTHM resonant reactions. The fully DCS at k bB → k bB ) is given by [19] d σd Ω bB d Ω sF dE sF = µ aA µ sF (2 π ) k sF ) k aA k bB ) µ bB (cid:12)(cid:12) M R (cid:12)(cid:12) , (58)where (cid:12)(cid:12) M R (cid:12)(cid:12) = 1 (cid:98) J a (cid:98) J A (cid:88) M B M b M s M a M A (cid:12)(cid:12) M M B M b M s ; M A M a ( k ˆk sF , k bB , k aA ) (cid:12)(cid:12) = 1ˆ J a ˆ J A (cid:88) M F M (cid:48) F M A M a M s M ( prior ) M F M s ; M A M a ( k sF ) ˆk sF , k aA ) [ M ( prior ) M (cid:48) F M s ; M A M a ( k sF ) ˆk sF , k aA )] ∗ × | N C | ( E − E sF ) + Γ / (cid:88) M B M b W M F M B M b ( k bB ) ) (cid:104) W M (cid:48) F M B M b ( k bB ) ) (cid:105) ∗ . (59)Here M M B M b M s ; M A M a ( k ˆk sF , k bB , k aA ) is the amplitude of the two-step THM reaction (16) with three particles inthe final state and M ( prior ) M F M s ; M A M a ( k sF ) ˆk sF , k aA ) is the amplitude of the transfer reaction (17), which is the first step4of the THM reaction, and is defined by Eq. (42). Term W M F M B M b ( k bB ) ) = √ π (cid:88) s bB l bB m sbB m lbB (cid:10) s bB m s bB l bB m l bB (cid:12)(cid:12) J F M F (cid:11)(cid:10) J b M b J B M B (cid:12)(cid:12) s bB m s bB (cid:11) Y l bB m lbB ( k bB ) ) × e i δ psbBlbBJF ( k bB ) ) (cid:115) µ bB Γ ( bB ) k bB ) . (60)represents the vertex form factor for the resonance decay F ∗ → b + B . The notations for spin-angular variableshave been introduced in section II.Taking into account that (cid:12)(cid:12) Γ[1 + i η ] (cid:12)(cid:12) = π η sinh( π η ) (61)we get for the Coulomb renormalization factor N C [12]: | N C ( E bB , ζ ) | = sinh[ π ( η sb + η sB )]sinh( πη sb ) sinh( πη sB ) πη sb η sB ( η sb + η sB ) × πη ζ sinh( πη ζ ) | F ( − iη sB , − iη sb , − | × exp (cid:20) ζ arctan 2( E bB ) − E bB )Γ (cid:21) . (62) C. Doubly DCS of transfer reaction populatingresonance state
It is convenient to integrate the fully DCS over Ω bB toget the doubly DCS, which is expressed in terms of theDCS of the reaction (17), corresponding to the first stepof the TH reaction. However, in the case under considera-tion, due to the presence of the Coulomb renormalizationfactor N C ( E bB , ζ ), the DCS obtained from integratingthe fully DCS over Ω bB cannot be expressed in terms ofthe DCS of the first step. The reason is that N C ( E bB , ζ )depends on the integration variable Ω bB . However, inthe following cases one can neglect this dependence:1. When | η sb | (cid:28) η sB ≈ η sF ) , where η sF ) = Z s Z F α µ sF /k sF ) , the imaginary partof η R can be neglected because of the narrowresonance. In this case, | N C | ≈ sF can be performed without anycomplications.2. When | η sb | (cid:28) m B (cid:29) m s , m b . By choos-ing the Galilean momenta k s = k sF and k bB asindependent variables, one can write k sB = m B Mm sB m bB k sF + m s m sB k bB ≈ k sF . (63)Then η sB = Z s Z B α µ sB /k sF and N C ( E bB , ζ )does not depend on k bB and integration over Ω k bB can be performed in a straightforward way. For simplicity, we assume that | N C ( E bB , ζ ) | = 1. Inte-grating the fully DCS over Ω bB and using the orthogo-nality of the spherical harmonics one obtains the doublyDCS: d σd Ω sF dE sF = 12 π Γ bB ( E bB ) − E bB ) + Γ / dσd Ω k sF , (64)where dσd Ω sF = µ aA µ sF π k sF ) k aA × (cid:88) M F M s M A M a | M ( prior ) M F M s ; M A M a ( k sF ) ˆk sF , k aA ) | (65)is the singly DCS of reaction (17). Note that integratingthe doubly DCS over E sF gives ∞ (cid:90) dE sF dσd Ω sF dE sF = Γ bB Γ dσd Ω sF , (66)where Γ bB is the partial resonance width for the decayof the resonance to the channel b + B . The reactionamplitude in Eq. (65) describes only transfer reactioninto the resonance without consideration of the resonancepropagator and decay into the final channel b + B . D. DWBA amplitude of reaction populatingresonance state
In the THM it is enough to consider the doubly DCS d σ/ ( d Ω k sF dE sF ) from which one needs to single out theastrophysical S ( E xA ) factor for the coupled two-channelresonant binary subreaction x + A → F ∗ → b + B (67)at E xA → E R ( xA ) , where E R ( xA ) = E xA ) − i Γ / x + A , S ( E xA ) E xA → E R ( xA ) = 5 · − πµ xA ˆ J F ˆ J x ˆ J A λ N m N e π η xA × Γ bB Γ xA (cid:0) E xA ) − E xA (cid:1) + Γ / , (68)where m N = 931 . λ N isthe nucleon Compton wave length. In this expression the5dimension of the S factor is MeV b. For weaker transi-tions we use units of keV b. Comparing Eqs. (68) and(64) one can observe that to single out the S ( E xA ) astro-physical factor from the latter it is enough to single outfrom the DCS dσ/d Ω sF the resonance width Γ xA . Thishas been done in section II. The derived transfer reactionamplitude M ( prior ) M F M s ; M A M a ( k sF ) ˆk sF , k aA ), according toEqs. (42), (39) and (33), is expressed in terms of √ Γ xA .In many cases the external part in Eq. (42) is smallcompared to the surface term and can be neglected. ThenEq. (64) can be rewritten as dσ T HM d Ω sF dE sF = S ( E xA ) e − π η xA P − l xA ( k (0) xA , R ch ) ˆ J x ˆ J A ˆ J F × ˆ l xA R ch π λ − N m − N (cid:12)(cid:12) W l xA (cid:12)(cid:12) dσ DW ZR ( prior ) d Ω sF . (69)We assigned to it the superscript “THM” because thisdoubly DCS can be used to analyze THM data. We as-sume that (cid:98) k xA ) is directed along the axis z , that is, Y l xA ,m lxA (cid:0)(cid:98) k xA ) (cid:1) = (cid:112) (2 l xA + 1) / π δ m lxA . With thisfor the DCS of reaction (17) populating the resonancestate F ∗ we get dσ DW ZR ( prior ) d Ω sF = µ aA µ sF π k sF k aA (cid:88) M F M s M A M a × (cid:12)(cid:12)(cid:12) M DW ZR ( prior ) M F M s ; M A M a ( k sF ) ˆk sF , k aA ) (cid:12)(cid:12)(cid:12) . (70) E. THM amplitude in plane-wave approach
In this part, we will simplify the equations derivedin the previous section using the plane-wave approach(PWA). The PWA, both in the initial and final states,a priori, cannot be applied for the analysis of the THMreactions. The reason is that the THM involves a colli-sion of charged particles a and A . To increase the DCSof the THM reaction, the E aA relative energy is selectedto be above the Coulomb barrier in the system a + A .But it cannot be too high because the resonance energies E xA ) measured using the THM kinematics, depend on E ( xA ) : the higher E ( xA ) the higher minimum resonanceenergy E xA ) that can be measured. Hence, for nucleiwith higher charges like in Ref. [13], the Coulomb dis-tortion in the initial state a + A cannot be neglected. Itwould be even more accurate to add nuclear distortion inthe initial state. However, taking into account the factthat in the THM, only the energy dependence of the DCSis measured, we assume that even such a drastic approx-imation as the PWA can be applied in some particularcases provided the PWA and DWBA excitation functionsare similar.The prior-form of the PWA follows immediately from a A x + F * a x F * A sss A FIG. 5. The diagrams describing the PWA transfer reactionamplitude a + A → s + F ∗ . The first diagram, which is thepole diagram, describes the simplest transfer reaction mech-anism of nucleus x . The second diagram is the exchange tri-angular diagram. The black filled sphere is the OFES s + A Coulomb+nuclear scattering amplitude in the Born approxi-mation generated by the potential V sA . the prior-form of the DWBA M P W A ( prior ) = (cid:10) Ψ (0) k sF ˜ φ R ( xA ) (cid:12)(cid:12) V xA + V sA (cid:12)(cid:12) ϕ sx Ψ (0) k aA (cid:11) . (71)The new notation here is the plane wave Ψ (0) k ij = e k ij · r ij describing the wave function of the relative motion of thenoninteracting particles i and j .Figure 5 shows the Feynman diagram representation ofthe amplitude M P W A ( prior ) . The amplitude of the firstdiagram has a peak at scattering angles of particle s nearthe forward direction. The second diagram in Fig. 5 isthe exchange triangular diagram. As a function of thescattering angle of the particle s its amplitude has twosingularities: due to the s + A Coulomb potential thetriangular diagram has the same singularity as the polediagram in Fig. 5. Including the triangular diagram leadsto the renormalization of the first diagram, see AppendixD from [20]. The second singularity of the amplitudecorresponding to the triangle diagram in the plane ofthe scattering angle of the particle s is generated by theskeleton triangle diagram (this singularity is generatedby the propagators of the triangle diagram) and leads tothe peak in the angular distribution at backward angles.As it has been done in the previous subsection, we splitthe PWA matrix element into the internal and externalparts to get M P W A ( prior ) = M P W A ( post ) int + M P W A ( prior ) S + M P W A ( prior ) ext . (72)6Here, M P W A ( post ) int = (cid:10) Ψ (0) k sF ˜ φ R ( xA ) (cid:12)(cid:12) V sx + V sA (cid:12)(cid:12) ϕ sx Ψ (0) k aA (cid:11) (73)is the post-form of the internal PWA amplitude, M P W A ( prior ) S = 12 e − i δ psxAlxAJF ( k xA ) ) (cid:115) µ xA k xA ) Γ xA × O l xA ( k xA ) R ch ) ϕ sx ( k sx ) W l xA i − l xA Y l xA ,m lxA ( (cid:98) k xA ) )(74)is the surface term of the prior-form PWA amplitude, k sx = k s − m s m a k a , k i is the OES momentum of the parti-cle i , ϕ a ( k sx ) is the Fourier transform of the s -wave ( sx )bound-state wave function ϕ a ( r sx ). Finally, the prior-form of the external PWA amplitude is written as M P W A ( prior ) ext = (cid:10) Ψ (0) k sF ˜ φ R ( xA ) (cid:12)(cid:12) V CxA + V CsA (cid:12)(cid:12) ϕ a Ψ (0) k aA (cid:11)(cid:12)(cid:12)(cid:12) r xA >R ch . (75)Taking into account that the distance between the parti-cles s and x is controlled by the bound-state wave func-tion ϕ a ( r sx ) for r xA > R ch , at large enough channel ra-dius R ch we replaced V sA in Eq. (75) by the Coulombpart V CsA . IV. APPLICATION OF DWBA FORMALISMFOR ANALYSIS OF RESONANT THMREACTION OF C + C FUSION
In this section we present a critical analysis of the ap-plication of the indirect Trojan Horse method (THM) tomeasure the astrophysical S ∗ factor of C + C fusion[13]. In the case under consideration, a = N, A = C, x = C, s = d , and F = Mg ∗ . Four different channelsin the final state were populated in the THM experi-ment: p + Na, p + Na (0.44 MeV), α + Ne, and α + Ne (1.63 MeV) [13]. The THM is a unique indirecttechnique that allows one to measure the astrophysicalfactors of the resonant reactions at low energies, wheredirect methods are not able to obtain data due to verysmall cross sections. The critical analysis of the THM inthis review is not aimed to taint the whole method whichdemonstrated its power in more than hundred publica-tions (see [6] and references therein). We critically reviewonly the analysis of the data in [13].The THM resonant reaction (16) involves two steps.The first step is transfer reaction (17) populating theresonance state F ∗ = x + A , and the second step isdecay of the resonance F ∗ → b + B . In other words, theTHM reaction is a process leading to three particles inthe final state. This makes analysis of such a reactionquite complicated. Special kinematical conditions should be fulfilled and angular correlation data areneeded to make sure that the reaction is dominantlycontributed by the resonant THM mechanism. Onlythen it will allow one to extract from the THM reactionan information about resonant binary reaction (15). In[13] the C( N , d )( α + Ne) and C( N , d )( p + Na)reactions were measured to obtain the S ∗ factors forthe carbon-carbon fusion and a sharp rise of the astro-physical S factor for carbon-carbon fusion determinedusing the indirect THM was reported. Here we outlinethe most important inconsistencies in the THM analy-sis and data from [13]. All the notations are given in [24].To analyze the measured data Tumino et al. [13] useda simple PWA developed by one of us (A.M.M.) ratherthan a generalized R -matrix approach based on thesurface-integral approach discussed in subsection III Dand originally published in Ref. [9]. The approach usesdistorted waves in both initial and final states. The PWAfollows from this more general approach when the dis-torted waves are replaced with the plane waves. ThePWA was successfully applied for analyses of many THMreactions in which the spectator is a neutron. It was alsoapplied to reactions at energies above the Coulomb bar-rier in the initial and final states, and when the interact-ing nuclei have small charges [3, 5, 6]. In the PWA it isassumed that the angular distribution of the spectator isforward-peaked in the center-of-mass system (quasi-freekinematics) and that the bound-state wave function ofthe spectator can be factorized out (see subsection III Dand Eq. (117) of Ref. [9] and Eq. (2) of Ref. [13]).Usage of the PWA can be justified only if the PWA andthe DWBA give similar energy dependence for the DCSof the transfer reaction. This is because in the THMonly the energy dependence of the astrophysical factor ismeasured while its absolute value is determined by nor-malizing the THM data to available direct data at higherenergies.Tumino et al. [13] reported that the astrophysi-cal S ∗ ( E ) factors extracted from the THM experimentdemonstrate a steep rise when the resonance energy E decreases. This rise would have profound implicationson different astrophysical scenarios as the carbon-carbonfusion rate calculated from the astrophysical S ∗ factorsdeduced in Ref. [13] significantly exceeds all previous es-timations of the reaction rate obtained by extrapolatingthe direct data to the low-energy region. For example,the reaction rate calculated in Ref. [13] at temperature T ∼ × K exceeds the adopted value [25, 26] by afactor of 500.The authors of Ref. [13] were rightly concerned aboutthe Coulomb barrier in the initial state. That is why inthe experiment the initial energy was above the Coulombbarrier. However, given the energy of the emitted parti-cles, neglecting the Coulomb effects in the final channelis unjustified. Below we present a detailed analysis ofthe TH resonant reactions based on the distorted-waveformalism. We take into account the distortions in the7initial, intermediate and final states.
A. Kinematics of the THM reaction
In Ref. [13] the normalization of the THM data to thedirect data was done in the energy interval E = 2 . − . E is the C − C relative kinetic energy.Here and in what follows we use E xA ≡ E . To checkwhether the PWA is justified, we consider the kinematicsof the THM in the energy interval covered by the THMexperiment [13]. In the experiment the relative N − Cenergy in the entrance channel was E aA = 13 .
845 MeV.From energy conservation in the THM reaction it followsthat E aA + Q = E f , where E f = E sF + E bB is the totalkinetic energy of the final three-body system s + b + B and Q = m a + m A − m s − m b − m B . From this equation weget that the total kinetic energy in the final d + p + Nachannel is E f = 5 . C − C relative energy E = 2 . C + C → p + Na, we have Q = 2 .
24 MeV, where Q = m x + m A − m b − m B . Accordingly, the energy in the p + Nachannel corresponding to E = 2 .
63 MeV is E p Na = 4 . p + Na system corresponding tothis energy is E d Mg = 0 .
93 MeV. This energy is wellbelow the Coulomb barrier in the d − Mg system, whichis about 3 MeV. Even on the lower end of the normaliza-tion interval corresponding to E = 2 . E d Mg = 1 .
06 MeV.At the energy of E = 1 . C − C chan-nel, which corresponds to the energy E p Na = 3 .
74 MeVin the exit channel, the relative energy E d Mg = 2 . .
56 MeV. This is due tothe fact that above 3 .
56 MeV, the resonance energy inthe p + Na channel is 3 .
56 + Q > . d − Mg relative energy is E d Mg <
0. Evenfor the energy of E = 0 .
805 MeV, which correspondsto E p Na = 3 .
05 MeV, the d − Mg relative energy is E d Mg = 2 .
75 MeV. The latter is close to but still belowthe Coulomb barrier. Thus we may conclude that theCoulomb interaction plays a very important role in theenergy interval exploited in [13] and, therefore, cannotbe neglected.
B. DWBA DCS
The presence of the strong Coulomb interaction forsuch deep sub-Coulomb processes in the final state ofthe transfer reaction significantly increases the DCS inthe backward hemisphere, shifting the peak of the angu-lar distribution of the deuterons to the backward angles.It completely contradicts to the PWA DCS in the c.m.
TABLE I. Parameters of the C − C potentials used tocalculate the resonance bin wave functions. E (MeV) No. V (MeV) r (fm) a (fm) width (MeV)2 . .
87 1 .
25 2 .
40 3 . × − . .
51 1 .
05 2 .
40 4 . × − . .
95 1 .
25 1 .
85 3 . × − . .
57 2 .
80 3 .
05 2 . × − . .
07 2 .
60 3 .
05 2 . × − . . .
80 2 .
30 1 . × − . .
47 4 .
50 4 .
50 2 . × − . .
284 4 .
20 4 .
50 2 . × − . .
798 4 .
50 4 .
04 9 . × − system, which has a pronounced peak at forward angles.Even at the lowest observed resonances at 0 . − . C-transferreaction are included.But what is even more important is the fact that thepresence of the strong Coulomb interaction significantlychanges the absolute values of the DCSs of the C trans-fer reaction and their energy dependence. The absolutevalues of the DCSs in the THM normalization interval be-come smaller than the corresponding PWA ones by morethan three orders of magnitude and they increase rapidlywhen the resonance energy decreases. That is one of themain reasons for the drop of the THM astrophysical fac-tors found in this work compared to those extracted in[13] using the PWA.Below we compare the PWA and DWBA DCSs. Toensure the convergence of the matrix elements for thetransfer reaction involving resonant states of Mg weuse the binned resonant wave functions. In the next threefigures we present three different curves corresponding tothree different potentials describing the resonances with l xA = 2 in Mg given in Table I. We use the standardnotations for the potential parameters shown in Table I: V is the depth of the Woods-Saxon potential, r and a arethe radial parameter and the diffuseness.The bin wave functions for the C − C resonancestates are generated using the potentials from Table Iand are obtained by integrating the C − C scatteringwave functions over the energy E with the interval widthof 0 . δ ( k C C )] exp [ − iδ ( k C C )] [22], where δ ( k C C ) is the C − C scattering phase shift. The binsizes affect the resulting bin wave functions, and, hence,the amplitude of the THM transfer reaction but they donot affect much the shapes of the angular distributions.8 − − − b i n w a v e f un c ti on (b) − − FIG. 6. (Color online) The bin wave functions calculatedfor three resonance energies E = 2 . , . . C − C system. Each panel contains three lines cor-responding to three different potentials for each resonanceenergy. The red solid, blue dashed and black dotted curvescorrespond to the potentials 1 , E = 2 . E = 1 . E = 0 . The resonance energies given in Table I are selectedfrom the high end, middle and low energy interval mea-sured in [13]. Note that we are not able to reproduceexactly the location of the resonances reported in [13]but the obtained resonance energy are pretty close tothe corresponding experimental ones. The bin wave func-tions for the three resonance energies constructed usingthe potentials from Table I are depicted in Fig. 6. Thehighly oscillatory behavior of the resonance wave func-tions is a clear evidence that the internal Coulomb orCoulomb+nuclear DWBA in the post form should besmall (see Section II B).The PWA DCSs for three resonance energies E =2 . , . . C − C system are shownin Fig. 7. Each panel contains three lines correspondingto three different potentials for each resonance energy,see Table I.Next we show the DWBA DCSs calculated using thebin wave functions shown in Fig. 6. Figure 8 presents theCoulomb DWBA DCSs calculated at the same three res- − −
0 30 60 90 120 150 180 θ c.m. (deg)(c)10 − − d σ / d Ω ( m b / s r) (b)10 − − − (a) FIG. 7. (Color online) The PWA DCSs for the N + C → d + Mg ∗ reaction at the relative kinetic energy E C N = 13 .
85 MeV populating three resonant states in Mg: E = 2 . , . . , E = 2 . E = 1 . E = 0 . onance energies of the system C − C. We performedcalculations using the pure Coulomb DWBA (thin lines)and the Coulomb + nuclear DWBA (thick lines). Theoptical-model potential parameters are taken from thecompilation [27], namely, parameters for the N − Cpotential at 27 . d − Mg potential at 3 . Mg subsystem depends on the excitationenergy of the latter. In principle, different optical po-tentials should be used in the exit channel for each Mgexcitation energy. However, our calculations suggest thatthe DCSs of the transfer reaction depend weakly on thechoice of the exit-channel optical model potentials. Thisis because the relative d + Mg energies in the exit chan-nel are so low that the Coulomb interaction dominatesover the exit-channel distorted waves. For this reason,the same exit-channel optical potential is used for all thecases.9 − − − x10 (a)10 − − d σ / d Ω ( m b / s r) (b)10 −
0 30 60 90 120 150 180 θ c.m. (deg)(c) FIG. 8. (Color online) The DWBA DCSs for the N+ C → d + Mg ∗ reaction at the relative kinetic energy E C N =13 .
85 MeV populating three resonant states in Mg. Panel(a): E = 2 . E = 1 . E = 0 . C − C bin wave functions for the potentials 1 , . First published in [24]. From the presented figures we can draw the followingconclusions: • The PWA and the DWBA DCSs differ significantlyboth in the angular distributions and energy de-pendences. In particular, the DWBA calculationsshow that in the interval of the resonance energies E = 1 . − . • The ratio of the DCSs from the PWA and theDWBA at E = 2 . . E = 2 . E = 0 . E = 2 . E decreases because the energy of the out-going deuteron increases as the Coulomb barrier isapproached.As we will see later [see Eq. (78) below for the S factor] the DWBA DCS appears in the denomina-tor. A very small DCS at high E should signifi-cantly increase the THM astrophysical factor. Asthe energy E decreases the DWBA DCS increasesand the S ( E ) factor quickly drops. For compar-ison we set our renormalization factor R ( E ) [seeEq. (80) below] equal to unity at E = 2 .
664 MeV,which is on the upper border of the THM normal-ization interval considered in [13]. The significantrise of the DWBA DCS toward small E is the fac-tor that most contributes to the drop of the THM S ( E ). C. Renormalization of THM astrophysical factors
Here we compare the astrophysical factors obtainedin the PWA and DWBA and obtain the renormaliza-tion factor of the THM S factor [13] taking into accountthe distortion effects in the initial and final states of thetransfer reaction.We start from the THM doubly DWBA DCS given byEq. (69) which can be rewritten asd σ THM d E dΩ k sF = K ( E ) S ( E ) (cid:12)(cid:12) W l xA (cid:12)(cid:12) dσ DW ZR ( prior ) d Ω k sF . (76)Here, K ( E ) = e − π η xA P − l xA ( k (0) aA , R ch ) ˆ J x ˆ J A ˆ J F ˆ l xA R ch π λ N m N (77)is a trivial kinematical factor, dσ DW ZR ( prior ) /d Ω k sF isthe zero-range DWBA cross section of the N + C → d + Mg ∗ reaction populating the isolated resonancestate, S ( E ) is the astrophysical factor.Correspondingly, the THM astrophysical factor deter-mined from Eq. (76) is S ( E ) = N F K ( E ) 1 (cid:12)(cid:12) W l xA (cid:12)(cid:12) d σ THM d E dΩ k sF × σ DWZR(prior) ( E, cos θ s ) / dΩ k sF . (78)0Here N F is an overall, energy-independent factor for nor-malization of the THM data to direct data, θ s is the scat-tering angle of the spectator s in the c.m. of the THMreaction. We recall that in the THM only the energydependence of the astrophysical factor is measured. Itsabsolute value is determined by normalizing the THM S ( E ) factor to the direct data available at higher ener-gies.Equations (76) and (78) are pivotal for understand-ing the challenges in extraction of the S ( E ) fac-tor from the THM DCS. Since in the normaliza-tion interval of E = 2 . − .
66 MeV the out-going deuterons are below the Coulomb barrier,d σ DWZR(prior) ( E, cos θ s ) / dΩ k sF is small and rapidly in-creases when the resonance energy E decreases. This in-crease of d σ DWZR(prior) ( E, cos θ s ) / dΩ k sF should reflect inthe behavior of d σ THM / (d E dΩ k sF ) and the THM S ( E )factor. As we already mentioned, in Ref. [13] a simplePWA was used instead of the distorted waves. The DCSas a function of E dF obtained using the PWA changesvery little compared to the change of the DWBA DCS.This is the main reason why the THM S ( E ) factors showunusually high rise when E decreases.In [13] the normalization interval was chosen to be E = 2 . − .
63 MeV. However, there are two resonanceswith negative parities that are questionable because thecollision of the two identical bosons C+ C cannot pop-ulate resonances with the negative parity. There are tworesonances with positive parities cited in [13]: at 2 . .
537 MeV. It was underscored in [13] that the THMdata reproduce the higher-lying resonance. That is whyhere we use the resonance at 2 .
664 MeV for the normal-ization of the THM data to direct ones. Thus, we assumehere that the normalization factor N F is determined bynormalizing the THM astrophysical factor to the directlymeasured resonance at E = 2 .
664 MeV. Practically weselected the normalization of the THM data on the edgeof the energy interval measured in [13].To find the renormalization of the THM astrophysi-cal factor presented in [13] we recall that in the PWAthe THM astrophysical factor for an isolated resonanceis given by S (PWA) ( E ) = N F K ( E ) d σ THM d E dΩ k sF ϕ a ( E ) |W l xA | . (79)The factor W l xA is given in Eq. (41) [9], ϕ a ( E ) is theFourier transform of the a = ( s x ) bound-state wavefunction. The Fourier transform, actually, depends onthe s − x relative momentum k sx , which in the caseunder consideration is k d C and expressed in termsof k d . From the energy conservation it follows that E sF = E aA − ε sx − E . Hence the Fourier transform ofthe bound-state wave function ϕ a ( k sx ) depends on E .By taking the ratio of the S ( E ) factors given by Eqs.(78) and (79) we get the renormalization factor for theTHM astrophysical factor presented in [13] due to theCoulomb (Coulomb + nuclear) distortions in the initial and final states of the THM transfer reaction: R ( E ) = ϕ a ( E ) ϕ a ( E N ) d σ DWZR(prior) ( E N , cos θ s ) / dΩ k sF d σ DWZR(prior) ( E, cos θ s ) / dΩ k sF , (80)where E N is the THM normalization energy. D. Astrophysical factors for the C − C fusionfrom THM reaction In this subsection we present new C − C fusion S ∗ = e . E S ( E ) factors obtained by renormalizingthe THM astrophysical factors presented in [13]. Forrenormalization the factor R ( E ) given in Eq. (80)is used. The DWBA DCSs are calculated using theFRESCO code [22]. For comparison we also calculated σ DWZR(prior) ( E, cos θ s ) / dΩ k sF including the nuclear dis-tortions. To calculate the optical-model distorted waveswe use the optical potentials from Ref. [27] as describedabove. Following [13] we use the normalization energy E N = 2 .
664 MeV.The energy dependence of the DWBAZR(prior) DCSat the deuteron scattering angle of 15 degree in the c.m.of the reaction is shown in Fig. 9. − e x c it a ti on f un c ti on E (MeV)
FIG. 9. (Color online) The excitation function of the THMtransfer reaction N + C → d + F ∗ calculated using theDWBAZR(prior) at the scattering angle of the deuteron of15 degree in the c.m. of the reaction. The solid red line is forthe pure Coulomb DWBAZR(prior) and the dashed blue linefor the Coulomb + nuclear DWBAZR(prior). First publishedin [24]. In Fig. 10 the total renormalized astrophysical factoris compared with original one from [13] . The total renor-malized astrophysical factor is R ( E ) S ∗ ( E ), where S ∗ ( E )is the total astrophysical factor taken from [13]. One cansee that at the resonance energies E = 0 . − . R ( E ) decreases the THM as-trophysical factors from [13] by about a factor of 10 .1 S f ac t o r ( M e V b ) E (MeV)
FIG. 10. (Color online) Total S ∗ ( E ) factors for C + Cfusion. Black solid line is the S ∗ ( E ) factor from [13]. The reddashed line is the renormalized R ( E ) S ∗ ( E ) factor calculatedusing the pure Coulomb distortions. The blue dotted lineis the renormalized R ( E ) S ∗ ( E ) factor calculated using theCoulomb plus nuclear distortions. First published in [24]. We conclude from Fig. 10 that the inclusion of the dis-torted waves in the initial and final states eliminates thesharp rise of the S ∗ ( E ) factors extracted in [13] using theTHM in the PWA. Our renormalized S ∗ factors do not,and are not supposed to, exhibit new resonances. Theyjust follow the resonance structure of the astrophysicalfactors obtained in [13].Our estimations of the DWBAZR(prior) DCSs of the C transfer reaction show that in the THM normaliza-tion interval of 2 . − .
664 MeV, the DWBA DCSs areof the order of 10 − − − mb/sr. Such small DCSs canhardly be measured in the coincidence experiment. Thatis why the THM data are not reliable at higher ener-gies. The absence in the THM data of a strong, isolatedresonance at E ∼ . E >
E. Concluding remarks
Concluding this section we emphasise once again thatthe THM is a powerful and unique indirect technique thatallows one to measure the astrophysical factors of the res-onant reactions at low energies, where direct methods arenot able to obtain data due to very small cross sections.However, we stress the following points.1. We question the validity of the results for the as-trophysical factors reported in [13] using the PWA.Since the THM deals with three-body reactionsrather than binary ones, a reliable theoretical anal-ysis of the THM data becomes critically important.For the THM reactions with the neutron-spectator or for the reactions with the energies above theCoulomb barrier and for interacting nuclei withsmall charges, the simple PWA works quite well andthe THM results are expected to be reliable. How-ever, this is not the case for the THM reaction un-der consideration, which aims to determine the as-trophysical factors of C + C fusion. In this pro-cess we deal with the strong Coulomb interactionsin the initial and final states of the THM transferreaction. The PWA follows from the more gen-eral distorted-wave-Born approximation (DWBA)in which the distorted waves are replaced by theplane waves and can be used only if the PWAcalculations provide a reasonable agreement withDWBA ones. It is not the case under consideration[24]. It has been demonstrated in [24] that the riseof the S ∗ factors at low energies seen in the afore-mentioned work was an artifact of using the PWA,which is not usable for the case under considera-tion. It was shown that such a rise disappears if theCoulomb (or Coulomb-nuclear) interactions in theinitial and final states are included. A very com-pelling evidence that the PWA should not be usedis presented in the pioneering work [29] where theexperimental angular distribution of the deuteronsfrom the THM reaction C( N , d ) Mg at 33 MeVincident energy of N. In this experiment the inci-dent energy was higher than in the THM exper-iment in [13]. Besides, the excited bound stateof Mg was populated rather than the resonancestate. All these facts make the energy of the outgo-ing deuterons higher than the Coulomb barrier inthe final d + Mg state (note that in the THM ex-periment the energies of deuterons were below theCoulomb barrier). Nevertheless, the experimentalangular distribution was flat and was perfectly re-produced by the DWBA calculation, which agreeswith the DWBA calculation in [24]. Moreover, itfollows from [29] that the momentum distributionof the deuterons disagrees with the one extracted in[13]. It casts doubt about the mechanism measuredin [13]. Note that the deuteron angular distributionwas not presented in [13]. The only criterium usedin the THM analysis to justify the PWA was thedeuteron momentum distribution. It worked forlighter nuclei [6] but not in the case under consid-eration in which different competing mechanismsdo contribute, such as B transfer from the Ctarget to N or Be transfer to C leaving Liin the resonance state decaying into d + α channel.The angular correlations of the final-state particles,which provide the most crucial information neededto identify the reaction mechanism [29], are missingin [13].2. Another evidence casting doubts on the energy de-pendence of the total S ∗ factor from [13] is indi-cated in [30]. This S ∗ factor exceeds significantlythe theoretical upper limit calculated in [31], see2Fig. 1 from [30].3. The THM doubly DCS in [13] does not correspondto the one described by the two-step THM mech-anism for the THM reaction N + C → d + Mg ∗ → α ( p ) + Ne( Na). The doubly DCS forthe THM mechanism is given by Eq. (39) from [24].Specifically, for the reaction under considerationdescribed by the THM mechanism, the energy ofthe outgoing deuterons corresponding to the Mgresonance energy at E = 2 . E d Mg = 1 . E = 2 . E d Mg = 0 .
97 MeV.Thus the deuterons are well below the Coulombbarrier of 3 MeV in the system d + Mg. Hence, theDWBA DCS dσ DW ZR ( prior ) /d Ω k sF of the transferreaction N + C → d + Mg ∗ , which is the firststep of the THM reaction, drops by two orders ofmagnitude on the interval 2 . − .
64 MeV inter-val what should lead to the decrease of the THMdoubly DCS. Hence, one can expect that as energy E increases the non-THM mechanisms, which arebackground, should dominate. These mechanismswere not identified (see the discussion above). Thatis why we doubt that the background did not con-tribute to the THM S ∗ factor at E > . S factors with theexperimental ones is another crucial issue. Thisdiscussion concerns a general matter of the applica-tion of the THM but is especially important for theapplication of the THM for heavier nuclei like thecase under consideration. The main advantage ofthe THM is the absence of the Coulomb-centrifugalbarrier in the entry channel of the C − C. Thatis why in the THM one can observe both low andhigh l xA resonances. There is the price one paysfor the absence of the barrier: the transferred par-ticle x = C in the THM reaction is off-the-energyshell. As the result in the doubly DCS appearsthe off-shell factor (cid:12)(cid:12) W l xA (cid:12)(cid:12) , see Eqs. (39) and(32) from [24]. This off-shell factor can increasethe contribution from high spin resonances, whichare suppressed in direct measurements. Fig. 11shows the off-shell factor (cid:12)(cid:12) W l xA (cid:12)(cid:12) calculated for the C( N , d ) Mg reaction at the incident energy of N 30 MeV and different l xA . The THM bringstwo modifications to the DCS of the binary reso-nant sub-reactions x + A → F ∗ → b + B . It removesthe penetrability factor P l xA in the entry channelof the sub-reaction and each partial wave is mul-tiplied by the off-shell factor (cid:12)(cid:12) W l xA (cid:12)(cid:12) , which maysignificantly modify the relative weight of the res-onances with different l xA . The selected channelradius R ch = 5 fm corresponds to the grazing colli-sion of two carbon nuclei. From the bottom panelof Fig. 11 one can conclude that the dominant con-tribution to the carbon-carbon fusion in direct mea- - 3 - 2 - 1 E ( M e V )
Off-shell factor
E ( M e V )
FIG. 11. (Color online) The upper panel: the off-shell factors (cid:12)(cid:12) W l xA (cid:12)(cid:12) for the THM reaction C( N , d ) Mg as functions ofthe C − C relative energy E at the channel radius R ch = 5fm calculated for 5 different C − C relative orbital angularmomenta l xA . Red dotted line - l xA = 0; magenta dottedline- l xA = 2; blue dashed-dotted line- l xA = 4; green dashed-dotted-dotted line- l xA = 6 and black solid line - l xA = 8. Thebottom panel: the ratio P l xA / ( E ) P ( E ) of the penetrabilityfactors calculated as function of the energy E calculated forthe channel radius R ch = 5 fm and different l xA ; P is thepenetrability factor for l xA = 0. The notations for the linesare the same as in the upper panel. First published in[30]. surements at low energies comes from two partialwaves: l xA = 0 and 2. It is also confirmed by calcu-lations in [32]. However, in the THM the dominantcontribution comes from high spins. This effect iscalled kinematical enhancement of higher spin res-onances. In particular, in [13] the resonance spinsup to 6 were assigned at energies below 1 . − assigned in [13] to the resonance at0 .
877 MeV is a mistake because a resonance withthe negative parity cannot be populated in the col-lision of two identical bosons such as C nuclei.The difference between the low-energy resonancespins contributed to the direct and indirect THMmeasurements should be taken into account whencomparing the direct and THM S ∗ factors.5. Despite of the criticism of the analysis in [13] weneed to acknowledge that a compelling evidence ofthe power of the THM at astrophysically-relevantenergies is also clearly demonstrated in [13] by dis-covering two strong resonances at 0 .
88 and 1 . . C − C fusionrates and will play the same role as the ”Hoyle”state [33] in the triple- α process of synthesis of C3[34]. Note that the models using the global po-tentials in [31, 35–37] can predict resonances onlyabove E = 2 . C+ C cluster resonance around E = 1 . Mg that would drastically enhance theenergy production and may provide a direct nu-clear driver for the superburst phenomenon. How-ever, no indication for such a state has not yet beenreported in direct measurements in which the min-imal measured energy is E ≈ . .
88 and 1 . C − C fusion research.Improving of the theoretical models, which will beable to predict the low-energy resonances detectedin the THM, is another imperative issue.6. Taking into account the critical importance of the C + C fusion in many astrophysical scenariostwo major challenges can be outlined: extendingdirect measurements below 2 . . S ∗ factors at en-ergies E < . S ∗ factors indeed contradict the ones from [13]. Thisis a practical confirmation of our claim that in thecase under consideration the PWA is not valid. V. THM FOR SUBTHRESHOLD RESONANCES
In the previous section we discussed the application ofthe THM for the resonant rearrangement reaction. An-other interesting extension of the THM is its applica-tion for the analysis of the rearrangement resonant re-actions proceeding through subthreshold resonances. Asubthreshold bound state (which is close to threshold)reveals itself as a subthreshold resonance in low-energyscattering or reactions. Subthreshold resonances playan important role in low-energy processes, in particular,in astrophysical reactions. In this section we considerthe application of the THM for analysis of the reactionsproceeding through a subthreshold resonance. To de-scribe the subthreshold resonance we use the R -matrixapproach. Some notations are modified compared to theones used in the previous sections.First, we present new R -matrix equations for the re-action amplitudes and astrophysical S factors for anal-ysis of reactions proceeding through subthreshold reso- nances. We consider elastic scattering and a resonantreaction for a subthreshold resonance coupled with openresonance channels for single and two-level cases. All theequations are expressed in terms of the formal and ob-servable reduced widths. The observable reduced widthis expressed in terms of the asymptotic normalizationcoefficient (ANC). We obtain an equation connecting theANC with the observable reduced width of the subthresh-old resonance, which is coupled with a resonance channel.Using a generalized R -matrix method and the surface-integral method we also derive equations for the THMreaction amplitude, fully and doubly DCSs in the pres-ence of the subthreshold state. We show that the THMcan incorporate equally well the subthreshold and realresonances. For more details see [41]. A. Single-channel single-level case
First we consider the single-level, single-channel R -matrix approach in the presence of the subthresholdbound state (also called subthreshold resonance). Theresonant elastic-scattering amplitude in the channel i = x + A with the partial wave l xA can be written in thestandard R -matrix form [10]: T ii = − i e − i δ hsi P i ( γ ( s ) i ) E − E i − [ S i ( E i ) − B i + i P i ] ( γ ( s ) i ) . (81)Here, γ ( s ) i is the reduced width amplitude of the sub-threshold bound state F s = ( x A ) ( s ) with the bind-ing energy ε ( s ) i = m x + m A − m F ( s ) , m j is the massof the particle j , E i ≡ E xA , E is the R -matrix en-ergy level, S i ( E i ) = R i Re (cid:104) d ln O l xA ( k i , r i ) / d r i (cid:12)(cid:12)(cid:12) r i = R i (cid:105) isthe R -matrix shift function in channel i , r i ≡ r xA isthe radius connecting centers-off-mass of the particlesin the channel i and k i ≡ k xA . B i ≡ B l xA is theenergy-independent R -matrix boundary-condition con-stant, P i ≡ P l xA ( E i , R i ) and R i ≡ R ch ( i ) are the pene-trability factor and the channel radius in the channel i , δ hsi ≡ δ hsxA l xA is the hard-sphere scattering phase shift inchannel i .If we choose the boundary-condition parameter B i = S i ( − ε ( s ) i ), in the low-energy region where the linear ap-proximation is valid, S i ( E i ) − S i ( − ε ( s ) i ) ≈ d S i ( E i )d E i (cid:12)(cid:12)(cid:12) E i = − ε ( s ) i ( E i + ε ( s ) i ) , (82)then at small E i T ii = 2 i e − i δ hsi P i (˜ γ ( s ) i ) ε ( s ) i + E i + i P i (˜ γ ( s ) i ) , (83)which has a pole at E i = − ε ( s ) i because P i vanishes for E i ≤
0. Here, ˜ γ ( s ) i is the observable reduced width of the4subthreshold resonance. The observable reduced width(˜ γ ( s ) i ) is related to the formal R -matrix reduced width( γ ( s ) i ) as(˜ γ ( s ) i ) = ( γ ( s ) i ) γ ( s ) i ) [d S i ( E i ) / d E i ] (cid:12)(cid:12) E i = − ε ( s ) i . (84)Equation (83) is of fundamental importance because itshows that a subthreshold bound state can be describedas a subthreshold resonance, that is, the resonance lo-cated at the negative energy − ε ( s ) i .Determining the ANC as the residue in the pole of thescattering amplitude corresponding to the bound-statepole, we get for the ANC of the subthreshold state [42][ C ( s ) i ] = 2 µ i R i (˜ γ ( s ) i ) W − η ( s ) i , l xA +1 / (2 κ ( s ) i R i ) , (85)where W − i η ( s ) i , l xA +1 / ( 2 κ ( s ) i R i ) is the Whittaker func-tion, η ( s ) i = Z x Z A /αµ i /κ ( s ) i and κ ( s ) i are the x − A Coulomb parameter and the bound-state wave numberof the subthreshold state F ( s ) , µ i is the reduced mass of x and A in the channel i , Z j e is the charge of nucleus j . Equation (85) relates the ANC and the the observablereduced width for single-channel case, but in the next sec-tion this equation will be generalized for the two-channelcase.The observable partial resonance width of the sub-threshold resonance is given by˜Γ ( s ) i ( E i ) = 2 P i (˜ γ ( s ) i ) = P i (cid:16) C ( s ) i W − η ( s ) i l xA +1 / (2 κ ( s ) i R i ) (cid:17) µ i R i . (86)Eq. (86) expresses the resonance width of the subthresh-old resonance in terms of the ANC and the Whittakerfunction of this subthreshold state taken at the channelradius R i . We have considered the connection betweenthe ANC and the reduced width amplitude for the boundstates. In the next subsection we consider the connectionbetween the ANC and the observable resonance width forreal resonances. B. Two-channel single-level case
Now we consider elastic scattering x + A → x + A in the presence of the subthreshold bound state F ( s ) inthe channel i = x + A which is coupled with the secondchannel f = b + B . The relative kinetic energies in thechannel i , E i ≡ E xA , and channel f , E f ≡ E bB , arerelated by E f ≡ E bB = E i + Q, Q = m x + m A − m b − m B > . (87) We assume that Q >
0, that is, the channel f is open for E i ≥ i = x + A in the single-level two-channel R -matrix approach is T ii = − i e − i δ hsi × P i ( γ ( s ) i ) E − E i − (cid:80) n = i,f [ S n ( E n ) − B n + i P n ] γ n , (88)where γ n is the formal reduced width in the channel n = i, f . Note that γ i ≡ γ ( s ) i . P n ≡ P l n ( E n , R n ) and R n are the penetrability factor and the channel radius in thechannel n . There are two fitting parameters, γ ( s ) i and γ f , in the single-level two-channel R -matrix fit at fixedchannel radii.Again, we use the boundary condition B n = S n ( − ε ( s ) i )and E = − ε ( s ) i . The energy in the channel i is E i = − ε ( s ) i and that in the channel f is E f = Q − ε ( s ) i . As-suming a linear energy dependence of S n ( E n ) at small E i , we get T ii ≈ i e − i δ hsi P i (cid:0) ˜ γ ( s ) i (cid:1) ε ( s ) i + E i + i (cid:80) n = i,f P n ˜ γ n , (89)where the observable reduced width in the channel n is˜ γ n = γ n (cid:80) t = i,f γ t [d S t ( E t ) / d E t ] (cid:12)(cid:12) E t = E ( s ) t , (90)again noticing that E ( s ) i = − ε ( s ) i and E ( s ) f = Q − ε ( s ) i .Correspondingly, the observable partial resonance widthin the channel n is ˜Γ n ( E n ) = 2 P n ˜ γ n , (91)with the total width ˜Γ( E i ) = ˜Γ ( s ) i ( E i ) + ˜Γ f ( E f ).The presence of the open channel coupled to theelastic-scattering channel generates an additional term n = f in the denominators of Eqs. (88), (89) and (90).Although the scattering amplitude vanishes at E i = 0 itcan be extrapolated to the bound-state pole bypassing E i = 0 using T ii E i →− ε ( s ) i ≈ κ ( s ) i ( − l xA +1 e i π η ( s ) i × R i W − η ( s ) i ,l xA +1 / (2 κ ( s ) i R i ) (˜ γ ( s ) i ) ε ( s ) i + E i + i P f ˜ γ f , (92)where P f = P f ( Q − ε ( s ) i , R f ). Again, the ANC, as aresidue in the pole of the scattering amplitude, is relatedto the observable reduced width (˜ γ ( s ) i ) of the subthresh-old state by Eq. (85), in which now (˜ γ ( s ) i ) is determined5by (˜ γ ( s ) i ) = ( γ ( s ) i ) (cid:80) t = i,f γ t [d S t ( E ) / d E ] (cid:12)(cid:12) E = E ( s ) . (93)The derivation of the connection between the ANC andthe reduced width of the subthreshold resonance in thepresence of an open channel f is a generalization of Eq.(85).It follows from Eq. (92) that in the presence of theopen channel f coupled with the channel i , the elastic-scattering amplitude has the bound-state pole shiftedinto the E i complex plane, i.e., E ( p ) i = − ε ( s ) i − i P f ( Q − ε ( s ) i , R f ) (˜ γ f ) .We have established a connection between the ANC,the observable reduced width and the observable reso-nance width (at E i >
0) of the subthreshold resonancestate. But, besides the subthreshold bound state in thechannel i , in Eq. (89) we have also a real resonance inthe channel f . In Refs. [42–44] the definition of the ANCwas also extended to real resonances. Here we remind theconnection between the resonance width, the ANC, andthe reduced width for the real resonance in the chan-nel f whose real part of the complex resonance energyis located at E (0) f . The ANC for the resonance state isdetermined as the amplitude of the outgoing resonancewave [44], which is the generalization of the definitionof the ANC for the bound state. For narrow resonancesthe ANC is expressed in terms of the resonance width as[42–44]: C f = µ f ˜Γ f ( E (0) f ) k (0) f e i [2 δ plf ( k (0) f ) − i π l f ] . (94)Here, C f is the ANC of the resonance state in the channel f , ˜Γ f ( E (0) f ) is the observable resonance width in the chan-nel f at the real part of the resonance energy, δ pl f ( k (0) f ) isthe non-resonant scattering phase shift in the channel f at the real part of the resonance momentum. C. S factor for reaction proceeding throughsubthreshold resonance in O In this subsection we present equations for the am-plitudes of reactions proceeding through a subthreshold resonance within the standard R -matrix equations gener-alized for the subthreshold state. Based on these ampli-tudes we obtain the corresponding astrophysical S factorswhich can be used to analyze experimental data obtainedfrom direct and indirect measurements. Note that herethe expressions for the astrophysical factors are writtenin a convenient R -matrix form and can be used by experi-mentalists for the analysis of similar reactions proceedingthrough subthreshold resonances.In section IV we considered the application of the THMfor the analysis of the binary resonant reaction in whichboth coupled channels, x + A and b + B were open. Herewe consider the resonant reaction x + A → F ( s ) → b + B (95)with Q >
0, proceeding through an intermediate reso-nance, which is a resonance in the exit channel f and thesubthreshold bound state F ( s ) = ( x A ) ( s ) in the initialchannel i . We assume also that Q − ε ( s ) i >
0, that is, thechannel f is open at the subthreshold bound-state polein the channel i .The single-level two-channel R -matrix amplitude de-scribing the resonant reaction in which in the initial statethe colliding particles x and A have a subthreshold boundstate and a resonance in the final channel f = b + B , canbe obtained by generalizing the corresponding equationsfrom Refs. [10, 45]: T f i = 2 i e − i ( δ hsi + δ hsf ) × (cid:112) P f γ f √ P i γ ( s ) i ε ( s ) i + E i + (cid:80) n = i,f [ S n ( E n ) − S n ( E ( s ) n ) + i P n ] γ n . (96)Here we remind that E ( s ) i = − ε ( s ) and E ( s ) f = Q − ε ( s ) .The astrophysical factor S ( E i ) is given by S ( E i )(keV b) = ˆ J F ( s ) ˆ J x ˆ J A λ N E N e π η i πµ i P f P i γ f ( γ ( s ) i ) (cid:16) ε ( s ) i + E i + (cid:80) n = i,f [ S n ( E n ) − S n ( E ( s ) n )] γ n (cid:17) + (cid:104) (cid:80) n = i,f P n γ n (cid:105) , (97)where J F ( s ) is the spin of the subthreshold state in the channel i = x + A , which is also the spin of the resonance6in the channel f = b + B , J j is the spin of the particle j , ˆ J = 2 J + 1, λ N = 0 . E N = 931 . / .Assume now that the low-energy binary reaction (95)is contributed by a few non-interfering levels. The sub-threshold resonance in the channel i = x + A , which iscoupled to the open channel f = b + B is attributed tothe first level, λ = 1, while other levels with λ > i and f of higherenergy levels E λ and spins J F λ . The astrophysical factor S ( E i ) is given by S ( E i )(keVb) = N (cid:88) λ =1 S λ ( E i )(keVb) , (98)where S λ ( E i )(keVb) = ν N E N πµ i e π η i ˆ J F ( λ ) ˆ J x ˆ J A P f λ P i λ γ f λ γ i λ (cid:16) E λ − E i − (cid:80) n = i,f [ S n λ ( E n ) − B n λ )] γ n λ (cid:17) + (cid:2) (cid:80) n = i,f P n λ γ n λ (cid:3) . (99)Here, all the quantities with the subscripts n, λ corre-spond to the channel n and level λ , γ i λ and γ f λ are thereduced width amplitudes of the resonance F ( λ ) in theinitial and final channels, γ i ≡ γ ( s ) i , E λ is the energylevel in the channel λ .Now we consider two interfering levels, λ = 1 and 2,and two channels in each level. All the quantities relatedto the levels λ = 1 and 2 have additional subscripts 1 or 2,respectively. We assume that the level λ = 1 correspondsto the subthreshold state in the channel i = x + A , whichdecays to a resonant state corresponding to the level λ =1 in the channel f = b + B . The level 2 describes theresonance in the channel x + A , which decays into theresonant state in the channel f = b + B . The level λ =2 lies higher than the level λ = 1 but these levels dointerfere. The reaction amplitude is given by T f i = − i e − i ( δ hsf + δ hsi ) (cid:112) P f (cid:112) P i (cid:88) λ τ γ f λ A λ τ γ i τ . (100)Here, A is the level matrix in the R -matrix method, (cid:0) A − (cid:1) λ τ =( − ε ( s ) i − E i ) δ λ τ − (cid:88) n = i,f γ n λ γ n τ × (cid:2) S n ( E n ) − S n ( E ( s ) n ) + i P n (cid:3) . (101)The corresponding astrophysical S ( E i ) factor is S ( E xA )(keVb) =20 π m N E N ˆ J F ( s ) ˆ J x ˆ J A µ i e π η i P f P i × (cid:12)(cid:12)(cid:12) (cid:88) λ τ γ f λ A λ τ γ i τ (cid:12)(cid:12)(cid:12) . (102) D. C( α, n ) O reaction In this subsection we present the THM analysis of theimportant astrophysical reaction C( α, n ) O, which isconsidered to be the main neutron supply to build upheavy elements from iron-peak seed nuclei in AGB stars.At temperature 0 . × K, the energy range wherethe C( α, n ) O reaction is most effective, the so-calledGamow window [26] is within ≈ −
230 keV withthe most effective energy at ≈
190 keV. This reactionwas studied using both direct and indirect (TH) meth-ods. Direct data, owing to the small penetrability fac-tor, were measured with reasonable accuracy down to E α C ≈
400 keV. Data in the interval 300 −
400 keVwere obtained with much larger uncertainty [46–50]. InRef. [50] the unprecedented accuracy of 4% was achievedat energies E α C >
600 keV. The dominant contribu-tion to the C( α, n ) O reaction at astrophysical ener-gies comes from the state O(1 / + , E x = 6356 ± E x is the excitation energy. Taking into accountthat the α − C threshold is located at 6359 . / + level is located at E α C = − ± O(1 / + ) wasadopted in the previous analyses of the direct measure-ments including the latest one in Ref. [52]. If this levelis a subthreshold bound state, then its reduced width isrelated to ANC of this level.However, in the recent paper [53] it has been deter-mined that this level is actually a resonance located at E α C = 4 . ± ± α of this resonance with l xA = 1 is negligibly small because it contains the pene-trability factor P . Hence, ˜Γ = ˜Γ n . The result obtainedin [53] is a very important achievement in the long his-tory of hunting for this near-threshold level. If this level7is actually a resonance located slightly above the thresh-old then the reduced width is related to the resonancepartial α width rather than to the ANC. Evidently thisresonance is not a Breit-Wigner type and it does notmake sense to use the ANC as characteristics of this res-onance.Here we present calculations of the astrophysical S fac-tors for the C( α, n ) O using the equations derivedabove. We fit the latest TH data [54] using assump-tions that the threshold level 1 / + is the subthresh-old state located at − . / + , l xA = 1 , E x = 6 . / − , l xA = 2 , E x = 7 .
165 MeV), ( 3 / + , l xA =1 , E x = 7 .
216 MeV), (5 / + , l xA = 3 , E x = 7 .
379 MeV)and (5 / − , l xA = 2 , E x = 7 .
382 MeV). Only two reso-nances, the second and the last one have the same quan-tum numbers and do interfere. Their interference can betaken into account using the S factor given by Eq. (102).For non-interfering resonances we use Eq. (98).In Fig. 12 we present the S factors contributed by fourdifferent resonant states located at E α C >
0. All theparameters of these resonances are taken from [52]. Weonly slightly modified the α -particle width of the wideresonance at E α C = 0 .
857 MeV taking it to be 0 . R i = 7 . α + C) and R f = 6 . n + O).As we see from Fig. 12, the contributions of all thenarrow resonances are negligible compared to the wideone (red solid line in Fig. 12). That is why we do not takeinto account the interference between two narrow 5 / − resonances. Thus eventually we can take into accountonly the wide resonance ( 3 / + , l xA = 1 , E x = 7 . / + , l xA = 1 , E x =6 .
356 MeV).
E. Threshold level / + , l xA = 1 , E x = 6 . MeV
Here we would like to discuss the threshold level E x =6 .
356 MeV. Until recently this level was considered tobe the subthreshold resonance located at E α C = − E α C = 4 . / + state dependson the reduced width in the entry channel α − C ofthe C( α, n ) O reaction and the reduced width in theexit channel n − O. The latter is determined with anacceptable accuracy, for example, in Refs. [52, 53]. Ifwe assume that the level E x = 6 .
356 MeV is subthresh-old resonance then its reduced width in the α -channelis expressed in terms of the ANC for the virtual decay O(1 / + , E α C = − → α + C. This ANC wasfound in a few papers [56–58]. The latest measurementof this ANC was published in [59]: ˜ C ( s ) α C = 3 . ± . E α C (MeV) S ( E α C ) [ M e V b ] FIG. 12. (Color online) The S factors for the C( α, n ) Oreaction as a function of the α − C relative kinetic en-ergy proceeding through four resonances: black dotted-dashed line -( 5 / − , l xA = 2 , E x = 7 .
165 MeV; solid redline- ( 3 / + , l xA = 1 , E x = 7 .
216 MeV); dashed brownline-(5 / + , l xA = 3 , E x = 7 .
379 MeV); dotted blue line-(5 / − , l xA = 2 , E x = 7 .
382 MeV). All the resonant parame-ters are taken from [52]. First published in [55]. fm − , which is the Coulomb renormalized ANC. Theproblem is that at very small binding energies the ANCof the subthreshold state becomes very large due to theCoulomb-centrifugal barrier. That is why in [17, 20] theCoulomb renormalized ANC was introduced in which theCoulomb-centrifugal factor was removed:˜ C = l xA !Γ(1 + l xA + η ( s ) ) C. (103)Here, Γ( x ) is the Gamma function, l xA is the orbitalangular momentum of the bound state. At small bindingenergies of the bound state, that is, at large η ( s ) , thefactor Γ(1 + l xA + η ( s ) ) becomes huge. Usually we areused to see the barrier factor to decrease the cross section,but here we see the opposite effect.However, in the R -matrix approach the quantity, whichwe need to calculate the astrophysical S factor, is thereduced width. From Eq. (85) one can express the ob-servable reduced width of the bound state in terms of theANC: ˜ γ = C W − η ( s ) , l xA +1 / (2 κ ( s ) i R i )2 µ R i . (104)The Coulomb-barrier factor, which significantly enhancesthe ANC, has an opposite effect on the Whittaker8function W − η ( s ) , l xA +1 / (2 κ ( s ) i R i ), so that the product C W − η ( s ) , l xA +1 / (2 κ ( s ) R ) is unaffected by the Coulomb-centrifugal barrier factor. It is convenient to rewrite Eq.(104) as ˜ γ = ˜ C ˜ W − η ( s ) , l xA +1 / (2 κ ( s ) i R i )2 µ i R i , (105)where µ i = µ α C , κ ( s ) i = (cid:113) µ i ε ( s ) i , ε ( s ) i = − W − η ( s ) , l xA +1 / (2 κ ( s ) R i ) = Γ(1 + l xA + η ( s ) ) l xA ! × W − η ( s ) , l xA +1 / (2 κ ( s ) R i ) . (106)For example, for the case under consideration, if thesubthreshold bound state is located at − l xA + η ( s ) ) = 2 . × for l xA = 1. For thechannel radius R i = 7 . W − η ( s ) , l xA +1 / (2 κ ( s ) R i ) =2 . × − while ˜ W − η ( s ) , l xA +1 / (2 κ ( s ) i R i ) = 0 . C W − η ( s ) , l xA +1 / (2 κ ( s ) i R i ) = ˜ C ˜ W − η ( s ) , l xA +1 / (2 κ ( s ) i R i )= 0 .
111 fm − / . (107)The reduced width changes very little if we assume thatthe threshold level 1 / + is a bound state. We used thesingle-particle α − C Woods-Saxon potential to generatethe bound-state wave function with the binding energy − r i >
0. Fol-lowing the R -matrix procedure, we accepted the internalregion as 0 ≤ r i ≤ R i , where R i = 5 . R i = 5 . − . R i = 4 .
93 fm. Thevalue of the single-particle reduced width decreased onlyby 2 .
5% compared to the value for the binding energyof − . − . / for the ANC ˜ C = 1 . − / and R i = 7 . . (cid:0) . − . (cid:1) keV / . Note that R i = 7 . F. Low-energy astrophysical factor for C( α, n ) O From Fig. 12 it is clear that only the experimental S factor generated by the broad resonance 3 / + , E α C = E α C (MeV) S ( E α C ) [ M e V b ] FIG. 13. (Color online) Astrophysical S factor for the C( α, n ) O reaction as a function of the α − C relative ki-netic energy. Square black boxes, solid green dots and shadedorange band are data from Refs. [48], [52] and [54], respec-tively. Red solid lines correspond to our calculations for the fitto the lower and upper limits of the TH data considering 1 / + state as − n = 158 . − and 4.7 fm − , respectively. Whereas,the blue dotted-dashed lines correspond to our calculationsfor the fit to TH data, considering 1 / + state as 4.7 keVthreshold resonance with Γ n = 136 keV [53] and the corre-sponding lower and upper values of observable reduced widthare 2.81 keV / and 3.6 keV / , respectively. For our calcu-lations we have used R α C = 7 . R n O = 6 . S factor. Firstpublished in [55]. .
857 MeV can be used for normalization of the TH dou-bly DCS at E α C > . W , see Eq. (41), was calculatedwithout the integral term in Eq. (41). Recalculating W taking into account the integral term we find that theTH results in [54] should be renormalized by 0 . S factor for the reaction C( α, n ) O.Our numerical values of the S (0) factors are:(1) for 1 / + , − n = 158 . S (0) =7 . +2 . − . × MeV b;(2) for 1 / + , 4.7 keV and Γ n = 136 keV [53], S (0) =7 . +2 . − . × MeV b.Thus, even the TH data, which provides the astro-9 a sA x F * F γ FIG. 14. Pole diagram describing the indirect radiative cap-ture reaction proceeding through the intermediate excitedstate F ∗ . physical factor at significantly lower energies than directmeasurements [52], cannot answer the question whetherthe threshold level is a subthreshold bound state or res-onance.In the analysis of the TH data, in the previous THpapers (see Refs. [54, 58]), only the two-stage mecha-nism proceeding through the intermediate threshold state1 / + has been taken into account. However, the single-step direct reaction C( α, n ) O also can contribute tothe low-energy cross section. Although the S factor of thedirect mechanism is flat and can be small, its interferencewith the two-stage resonant mechanism can change thetotal S factor. However, the accuracy of the existing datadoes not allow us to determine the contribution of the di-rect mechanism.
VI. THM FOR RADIATIVE CAPTUREREACTIONSA. Introduction
The THM can be used to determine different reactionsproceeding through the resonance. In the previous sec-tions we considered the THM for resonant rearrangementreactions. In this section we focus on the applicationof the THM to measure resonant radiative-capture re-actions at energies so low that direct measurements canhardly be performed due to the negligibly small pene-trability factor in the entry channel of the reaction. Wepresent here the theory of the indirect THM to treatthe resonant radiative-capture processes when only a fewsubthreshold bound states and resonances are involved,and statistical methods cannot be applied. As in the pre-vious sections, the developed formalism is based on thegeneralized multi-level R -matrix approach and surface-integral formulation of the transfer reactions, which arethe first stage of the indirect reaction mechanism de-scribed by the diagram depicted in Fig. 14 [60]. Thedeveloped formalism allows one to study the photon’sangular distribution correlated with the scattering angleof one of the final nuclei formed in the transfer reaction.There are many papers devoted to the angular corre-lation of the photons emitted in nuclear transfer reac-tions with final nuclei, see, for example, [61] and refer-ences therein. The approach, which is outlined below, can be applied for the C( Li , d γ ) O reaction, whichcan provide important information about the astrophys-ical C( α, γ ) O process. This astrophysical reaction iscontributed by two interfering subthreshold resonances([1], sect. 4.5). The subthreshold bound state may re-veal itself as a resonance in the case of the radiative cap-ture, which can occur to the wing of the subthresholdstate at positive energy forming the intermediate excitedstate. The excited bound state subsequently decays tolower lying states by emitting a photon. In this case, thesubthreshold bound state is characterized by a resonancewidth in complete analogy with the real resonance [42].Numerous attempts to obtain the astrophysical fac-tor of the C( α, γ ) O reaction, both experimental andtheoretical, have been made for almost 50 years [62–98].This reaction is contributed by interfering E E E − resonance at − .
045 MeV with the low-energy tail of the resonance 1 − , E α C = 2 .
423 MeV,where E α C is the α − C relative kinetic energy. The E O through the wing of the subthreshold boundstate 2 + , E α C = − .
245 MeV. In addition, to fit the ex-perimental data, usually a few artificial levels are addedto fit E E E − , E α C = 2 .
423 MeV the cross section is only about40 −
50 nb [92–94] . Moreover , the E − states to the ground state of O is isospin forbiddenfor the T = 0 component and is possible only due to asmall admixture of the T = 1 components.Extremely small penetrability factor at E α C ≤ C( α, γ ) O reaction at en-ergies E α C ≤ E α C ∼ . ≤ C( α, γ ) O reac-tion down to 1 MeV.The THM allows one not only to determine the astro-physical S factor down to energies E α C ∼ . E A ( a, s γ ) F proceeding through the subthreshold and realresonances.To measure the cross section of the binary process x + A → F ∗ → γ + F (108)proceeding through the intermediate resonance F ∗ at as-0trophysical energies we suggest to measure the surrogatereaction [two-body to three-body process (2 → a + A → s + F ∗ → s + γ + F (109)in the vicinity of the quasi-free kinematics region [3]. It isassumed that the incident particle a = ( sx ) is acceleratedat energies above the Coulomb barrier. The first stage ofthe THM reaction is the transfer reaction a + A → s + F ∗ populating the wing of the subthreshold bound state at E xA > F ∗ decays to the ground state F by emitting a photon.Physics of the THM has already been discussed in theprevious sections in which the expression for the ampli-tude of the transfer reaction (16) in the surface-integralapproach and the DWBA was derived. We assume thatthe energies of the particles in the initial and final statesin reaction (109) are above the Coulomb barrier. Inthis case, in contrast to the N + C one in whichthe deuterons were below the Coulomb barrier, it makessense to use the PWA, see Section III E and [3]. The com-parison of the PWA and DWBA has been done in manyTHM papers [3, 99–101]. In these papers, the momentumdistribution of the spectator was calculated in the PWAand DWBA. Both calculations agreed with each otherand experimental data within the range of the QF peak.The most detailed comparison of the PWA and DWBAwas done in [101]. It was confirmed that the angulardistributions of the spectator calculated in the DWBAand the plane-wave impulse approximation agree quitewell within the QF peak. The differences in the ratiosof the integrated resonance cross sections calculated inthe PWA and DWBA are less than 19%, compared withthe experimental uncertainties. Therefore, when no ab-solute values of the cross sections are extracted, the PWAdescription is more preferable than DWBA because theformer does not depend on the optical potentials whichfor light nuclei are not known accurately at low energies.
B. Amplitude of indirect resonantradiative-capture reaction
Let us consider the radiative capture reaction (108)proceeding through the wing (at E xA >
0) of thesubthreshold bound state (aka subthreshold resonance) F ∗ = F ( s ) , where F ( s ) = ( x A ) ( s ) , or a real resonanceat E xA >
0. We assume that both can decay to theground state F = ( xA ). To measure the cross section ofthis reaction at astrophysically relevant energies wheresubthreshold resonances can be important (for the rea-sons explained above), we use the indirect reaction (109).First, we derive the reaction amplitude of the indirectradiative-capture process and then the fully DCS of re-action (109). After that, by integrating over the anglesof the emitted photons, we get the doubly DCS. The interference of the subthreshold bound state and the res-onance, which both decay to the ground state F = ( xA ),is taken into account. Cleary, this case can be applied forthe E E C( α, γ ) O.To describe the radiative capture to the ground statethrough two interfering states we use the single-channeltwo-level generalized R -matrix equations developed forthe three-body reactions 2 particles → particles [3, 9].We also take into account the interference of transitionswith different multipolarities L . Thus, we take into ac-count the interference of radiative decays from differentlevels with the same multipolarity and interference oftransitions from various levels with different multipolar-ities.The indirect reaction described by the diagram shownin Fig. 14 proceeds as a two stage process. The first stageis transfer of particle x (stripping process) to the excitedstate F τ , τ = 1 ,
2, where F = F ( s ) is the subthresholdresonance and F is the resonance state at E xA >
0. Nogamma is emitted during the first stage. On the secondstage the excited state F τ decays to the ground state F = ( xA ) by emitting a photon. Then the indirect re-action amplitude followed by the photon emission fromthe intermediate subthreshold resonance and resonancetakes the form: M M s M M F M a M A = (cid:88) τ,ν =1 (cid:88) M F ( s ) ν M F ( s ) τ V M F MM F ( s ) ν ν A ν τ M M s M F ( s ) τ M a M A τ . (110)Here V ν ≡ V M F MM F ( s ) ν ν is the amplitude of the radiative de-cay of the excited state F ν ( ν = 1 ,
2) to the ground state F = ( xA ), A ν τ is the matrix element of the level ma-trix in the R -matrix method, M i is the projection of thespin J i of particle i , M F ( s ) τ is the projection of the spin J F ( s ) of the subthreshold resonance ( τ = 1) and reso-nance ( τ = 2), M is the projection of the angular momen-tum of the emitted photon. Also number 2 for the upperlimit of the first sum represents the number of the levelsincluded. We assume that the spins of the subthresholdresonance and real resonance are equal F = F = F ( s ) and these resonances do interfere. At the moment weconfine ourselves by transition with one multipolarity L .That is why the index L is omitted. Later on we takeinto account transitions with different L . M M s M F ( s ) τ M a M A τ isthe amplitude of the direct transfer reaction a + A → s + F τ , (111)populating the intermediate excited state F τ . The reac-tion (111) is the first stage of the indirect reaction (16).Below we present the general equations for thefully and doubly differential cross sections of the indi-rect radiative-capture reactions proceeding through sub-threshold and isolated resonances.1 C. Triply DCS
Let us consider the indirect resonant reaction (109)contributed by different interfering multipoles L . Foreach L we assume a two-level contribution. The deriva- tion of the amplitude of this reaction is given AppendixB. Then the fully DCS of the resonant indirect radiative-capture reaction for unpolarized initial and final particles(including the photon) in the center-of-mass of the reac-tion (109) is given byd σ dΩ s dΩ γ d E sF = µ aA µ sF ˆ J a ˆ J A (2 π ) k sF k γ k aA (cid:88) M a M A M s M F M λ (cid:12)(cid:12)(cid:12) M M s M F M λM a M A (cid:12)(cid:12)(cid:12) = − π ) µ aA µ sF ˆ J x ˆ J A ϕ a ( p sx ) R xA µ xA k sF k aA ( − J F − j xA (cid:88) L (cid:48) L ( − L (cid:48) + L k L (cid:48) + L +1 γ ˆ J L (cid:48) F ( s ) ˆ J LF ( s ) (cid:112) ˆ L (cid:48) ˆ L × (cid:88) l (cid:48) xA l xA l i L (cid:48) − l (cid:48) xA − L + l xA (cid:113) ˆ l (cid:48) xA ˆ l xA ˜ W ∗ l (cid:48) xA ˜ W l xA (cid:40) j xA l (cid:48) xA J L (cid:48) F ( s ) l J LF ( s ) l xA (cid:41) (cid:40) J L (cid:48) F ( s ) J F L (cid:48) L l J LF ( s ) (cid:41) × γ τ (cid:48) j xA l (cid:48) xA J L (cid:48) F ( s ) γ τ j xA l xA J LF ( s ) (cid:88) ν (cid:48) , ν, τ (cid:48) , τ =1 (cid:2) γ J L (cid:48) F ( s ) ( γ ) ν (cid:48) J F L (cid:48) (cid:3) ∗ [ γ J LF ( s ) ( γ ) ν J F L ] (cid:2) A L (cid:48) v (cid:48) τ (cid:48) (cid:3) ∗ (cid:2) A Lν τ (cid:3) × (cid:10) l (cid:48) xA l xA (cid:12)(cid:12) l (cid:11) (cid:10) L (cid:48) L − (cid:12)(cid:12) l (cid:11) [1 + ( − L (cid:48) + L + l ] P l ( cosθ ) . (112)To obtain Eq. (112) we adopted z || ˆp xA , that is, Y l m l ( ˆp xA ) = ˆ l √ π δ m l . Thus, in the PWA, the direc-tion ˆp xA becomes the axis of the symmetry. Note thatif we replace the plane waves by the distorted waves, thevestige of this symmetry will still survive [61].We remind that the radiative transition J LF ( s ) → J F is the electric EL where J LF ( s ) is the spin of the intermediatestate (subthreshold resonance or resonance).For a more simple case when only one multipole L contributes into the radiative transition, the fully DCStakes the form:d σ dΩ s dΩ γ d E sF = − π ) µ aA µ sF ˆ J x ˆ J A ϕ a ( p sx ) R xA µ xA k sF k aA k ˆ Lγ ( − J F − j xA ˆ L ( ˆ J LF ( s ) ) (cid:88) l xA l ˆ l xA × (cid:12)(cid:12) ˜ W l xA (cid:12)(cid:12) (cid:40) j xA l xA J LF ( s ) l J LF ( s ) l xA (cid:41) (cid:40) J LF ( s ) J F LL l J LF ( s ) (cid:41) (cid:88) ν (cid:48) , ν, τ (cid:48) , τ =1 (cid:2) γ J LF ( s ) ( γ ) ν (cid:48) J F L (cid:3) ∗ [ γ J LF ( s ) ( γ ) ν J F L ] (cid:2) A Lv (cid:48) τ (cid:48) (cid:3) ∗ (cid:2) A Lν τ (cid:3) × , γ τ (cid:48) j xA l xA J LF ( s ) γ τ j xA l xA J LF ( s ) (cid:10) l xA l xA (cid:12)(cid:12) l (cid:11) (cid:10) L L − (cid:12)(cid:12) l (cid:11) P l ( cosθ ) . (113)The off-shell factor ˜ W l xA is defined in Appendix B, seeEq. (144) Though we formally keep the summation over l xA , in the long-wavelength approximation for given L at astrophysically relevant energies only minimal allowed j xA contributes.The fully DCS depends on k s and k γ . Because weneglected the recoil of the final nucleus F , k s and k γ arerelated by Eq. (152). We remind that we selected axis z || p xA . Hence the photon’s scattering angle is countedfrom p xA , which itself is determined by k s . Thus theangular dependence of the fully DCS determines the an-gular correlation between the emitted photons from theintermediate excited state F ∗ and the spectator s . Since we consider the three-body reaction (16), the angular cor-relation function also depends on the spins J LF ( s ) of theintermediate nucleus F ∗ which decays to F .We note that(i) The most important feature of the indirect reactionfully DCS is that it does not contain the penetrabilityfactor P l xA ( E xA , R xA ) in the entry channel of the sub-reaction (95). This factor is the main obstacle to mea-sure the astrophysical factor of this reaction if one usesdirect measurements. The absence of this penetrabilityfactor in the entry channel of the sub-reaction allows oneto use the indirect method to get the information aboutthe astrophysical factor of the sub-reaction.2(ii) The indirect reaction fully DCS is parameterized interms of the formal R -matrix width amplitudes, whichare connected to the observable resonance widths.(iii) The final expression for the indirect reaction triplyDCS does not depend on the R -matrix hard-sphere scat-tering phase shift.By choosing QF kinematics, p sx = 0, one can providethe maximum of the fully DCS due to the maximumof ϕ a ( p sx ). At fixed k s the fully DCS determinesthe emitted photon’s angular distribution, which iscontributed by different interfering multipoles L . Bymeasuring the photon’s angular distributions at differentphoton’s energies (that is, at different k s or E xA ) one can determine the energy dependence of the photon’sangular distribution. However, a wide variation of k s away from the QF kinematics p sx = 0 will decrease theDCS due to the drop of ϕ a ( p sx ). Usually, in indirectmethods k s is varied in the interval in which p sx ≤ κ sx [3]. D. Doubly DCS
Integrating the fully DCS over the the photon’s solidangle Ω γ we get the non-coherent sum of the doublyDCSs with different multipoles L :d σ dΩ s d E sF = 1(2 π ) µ aA µ sF ˆ J x ˆ J A ϕ a ( p sx ) R xA µ xA k sF k aA (cid:88) L (cid:113) ˆ L ˆ J LF ( s ) k ˆ Lγ (cid:88) l xA (cid:12)(cid:12) ˜ W l xA (cid:12)(cid:12) × (cid:88) ν (cid:48) , ν, τ (cid:48) , τ =1 (cid:2) γ J LF ( s ) ( γ ) ν (cid:48) J F L (cid:3) ∗ [ γ J LF ( s ) ( γ ) ν J F L ] (cid:2) A Lv (cid:48) τ (cid:48) (cid:3) ∗ (cid:2) A Lν τ (cid:3) γ τ (cid:48) j xA l xA J LF ( s ) γ τ j xA l xA J LF ( s ) . (114)Despite of the virtual transferred particle x in the diagram of Fig. 14, using the surface-integral approach and thegeneralized R -matrix we can rewrite the doubly DCS in terms of the OES astrophysical factor for the resonant radiativecapture A ( x, γ ) F for the electric transition of the multipolarity L and the relative orbital angular momentum l xA ofparticles x and A in the entry channel of the A ( x, γ ) F radiative capture. In the R -matrix formalism this astrophysicalfactor is given by [102] S ELl xA ( E )( M eV b ) = 2 π λ N ˆ J LF ( s ) ˆ J x ˆ J A µ xA m N e π η xA P l xA ( E, R xA ) 10 − k ˆ Lγ (cid:12)(cid:12)(cid:12) (cid:88) ν,τ =1 [ γ J LF ( s ) ( γ ) ν J F L ] (cid:2) A Lν τ (cid:3) γ τ j xA l xA J LF ( s ) (cid:12)(cid:12)(cid:12) . (115)Here, µ xA is the x − A reduced mass expressed in MeV, η xA is the x − A Coulomb parameter at their relative energy E ≡ E xA . Then the indirect doubly DCS takes the form:d σ dΩ s d E sF = K ( E ) N F ϕ a ( p sx ) R xA (cid:88) L (cid:118)(cid:117)(cid:117)(cid:116) ˆ L ˆ J LF ( s ) (cid:88) l xA e − π η xA (cid:12)(cid:12) ˜ W l xA (cid:12)(cid:12) P l xA ( E, R xA ) S ELl xA ( E ) , (116)where K ( E ) = 10 (2 π ) µ aA µ sF m N λ N k sF k aA (117)is the kinematical factor, N F is an energy-independentTHM normalization factor. To determine the astrophys-ical factor from the indirect doubly DCS we need to iden-tify the region where accurate direct data are availableand only one resonance dominates with given L and l xA .By normalizing in this region the astrophysical factor ob-tained from the indirect measurement to the experimen- tal one we get S ELl xA ( E ) = N F d σ dΩ ˆk s d E sF (cid:115) ˆ J LF ( s ) ˆ L KF ϕ a ( p sx ) R xA × e π η xA P l xA ( E, R xA ) (cid:12)(cid:12) ˜ W l xA (cid:12)(cid:12) − . (118)Using this normalization factor we can determine the as-trophysical factors at energies E xA → x − A energies covering the interval from low energies relevantto nuclear astrophysics up to higher energies at whichdirect data are available. To cover a broad energy rangeat fixed energy of the projectile, the energy, and scatter-ing angle of the spectator should be varied near the QFkinematics ( p sx = 0).(2) Obtaining the indirect doubly DCS by integrating thefully DCS over the photon’s scattering angle.(3) Expressing the astrophysical factor in terms of theindirect doubly DCS.(4) Normalization of astrophysical factor to the availableexperimental data at higher energy.(5) Determination of the astrophysical factor at astro-physical energies. VII. RADIATIVE CAPTURE C( α, γ ) O VIAINDIRECT REACTION C( Li , d γ ) O A. Introduction
In this section we discuss the practical application ofthe developed formalism for the analysis of the indirectreaction C( Li , d γ ) O to obtain the information aboutthe astrophysical factor for the C( α, γ ) O at energies < L = 1 and L = 2 elec-tric transitions [72, 75, 77, 80, 85]. E O with J F = 0 and zero α − C or-bital angular momentum proceeds as resonant capturethrough the wing at E α C > − at E α C = − .
045 MeV, which works asthe subthreshold resonance. Besides, the E − resonance locatedat E R = 2 .
423 MeV. The E + state at E α C = − . + resonance at 2 .
68 MeV.These four states are observable physical states con- tributing to the low-energy radiative capture under con-sideration. In addition to these states, when fitting thedata an artificial level was added for E E α C = 0 . E E J = 1 , E α C = − .
045 MeV and the resonance J = 1 , E R = 2 .
423 MeV cannot decay by the E O because all of them haveisospin T = 0. Note that the observed weak E J = 1 , T = 0 states is possible only dueto the small admixture of the higher lying J = 1 , T = 1states [67].Let us estimate the recoil effect of the nucleus F forthe THM reaction C( Li , d γ ) O at the most effectiveastrophysical energy E xA = E α C = 0 . k γ ≈ E aA = 7MeV. As we will see below (Fig. 17) at 0 . θ = 52 ◦ ,where θ is the angle between p α C and k γ . In the QFkinematics p α C || k d , where k s = k d , that is, θ (cid:48) = θ . At θ = 52 ◦ , which is the maximum of the photon’s angulardistribution and is close to the maximum of the angu-lar distribution for the E ∼ . E ◦ where the recoileffect vanishes.The reduced widths of the subthreshold resonances areknown from the experimental ANCs [59, 75] and the re-duced width of the 1 − , .
423 MeV resonance is deter-mined from the resonance width. We disregard the cas-cade transitions to the ground state of O through sub-threshold states. According to [67], the sum of all cascadetransitions contributes only 7 − E O.For the case under consideration J x = 0 , J A =0 , j xA = 0 , l xA = L = J LF ( s ) , J F = 0. Hence, the expres-sion for the fully DCS for the case under considerationsimplifies to4d σ dΩ s dΩ γ d E sF = − µ aA µ sF (2 π ) ϕ a ( p sx ) R xA µ xA k sF k aA (cid:88) L (cid:48) L ( − L (cid:48) + L k L (cid:48) + L +1 γ (cid:112) ˆ L (cid:48) ˆ L × ˜ W ∗ L (cid:48) ˜ W L γ τ (cid:48) L (cid:48) L (cid:48) γ τ L L (cid:88) l (cid:10) L (cid:48) L (cid:12)(cid:12) l (cid:11) (cid:10) L (cid:48) L − (cid:12)(cid:12) l (cid:11) P l (cos θ ) × (cid:88) ν (cid:48) , ν, τ (cid:48) , τ =1 (cid:2) γ L (cid:48) ( γ ) ν (cid:48) L (cid:48) (cid:3) ∗ [ γ L ( γ ) ν L ] (cid:2) A L (cid:48) v (cid:48) τ (cid:48) (cid:3) ∗ (cid:2) A Lν τ (cid:3) (119)Here, a = Li , A = C , s = d, x = α, F = O. Thisexpression is used for the analysis of the indirect reaction C( Li , d γ ) O at low energies. We outline below somedetails of the calculations.After integration over the photon’s solid angle we getthe indirect doubly DCS (116) in which l xA = L . Then atenergies near the 1 − resonance at 2 .
423 MeV where, as wewill see below, the E S E ( E xA ) = N F d σ dΩ ˆk s d E sF K ( E xA ) ϕ a ( p sx ) R xA × e π η xA P ( E xA , R xA ) (cid:12)(cid:12) ˜ W (cid:12)(cid:12) − . (120)The S E astrophysical factor was measured at energiesnear 2 .
423 MeV with a very good accuracy [67, 80, 84].Should we have the experimental indirect doubly DCSexpressed in arbitrary units, we can use Eq. (120)to normalize the S E ( E xA ) to the experimental one athigher energies. After that, having indirectly measuredthe doubly DCS at 0 . S E (0 . S E (0 . E α C energies for the C( α, d γ ) O reactionand study how it is affected by the interference char-acter (constructive or destructive) of the 1 − subthresh-old bound state and 1 − resonance. In the R -matrixapproach the fitting parameters are the formal reducedwidths γ τ j xA l xA J F ( s ) which are related to the observableones ˜ γ τ j xA l xA J F ( s ) by˜ γ τ j xA l xA J F ( s ) = γ τ j xA l xA J F ( s ) γ τ j xA l xA J F ( s ) [d S l xA ( E xA ) / d E xA ] (cid:12)(cid:12) E xA = E τ . (121) R -matrix energy levels, E τ , are E = − ε ( s ) xA and E = E R , where ε ( s ) xA is the binding energy of the subthresh-old bound state ( xA ) ( s ) and E R is the resonance en-ergy corresponding to the level τ = 2. The observablereduced widths (˜ γ ) and (˜ γ ) are expressed interms of the corresponding ANCs of the subthresholdbound states by Eq. (85). For the ANCs of the 1 − and 2 + subthreshold states we adopted [ C ( s )( α C)1 ] = 4 . × fm − and [ C ( s )( α C)2 ] = 1 . × fm − [59], respec-tively. In all the calculations, following [75], we use the channel radius R ch ( α C) = 6 . − is expressed in termsof the observable resonance width of this resonance (seeEq. (91)). For this resonance we adopt ˜Γ = 0 . E − subthreshold bound state, that is, E = − ε ( s ) xA (1) . For the E + subthreshold bound state E = − . γ γ ) 1 0 1 , γ γ ) 2 0 1 and γ γ ) 1 0 2 in terms of corre-sponding observable reduced widths, which are related tothe observable radiative resonance widths by Eq. (150)[10].Another important point to discuss is the kinematicsof the indirect reaction. The fully DCS is proportional to ϕ d α ( p d α ), which is shown in Fig. 15. Here ϕ Li ( p d α ) ≡ ϕ d α ( p d α ). The maximum of ϕ d α ( p d α ) at p d α = 0 (QFkinematics) also provides the maximum of the fully DCS.Here p d α is the d − α relative momentum in the three-rayvertex Li → d + α of the diagram in Fig. 14.To calculate the Fourier transform of the Li = ( d α )bound-state wave function we use the Woods-Saxon po-tential with the depth V = 60 . r = r C = 1 .
25 fm and diffuseness a = 0 .
65 fm. Thispotential provides the d − α bound state with the bind-ing energy ε d α = 1 .
474 MeV [103] . The correspond-ing bound-state wave number of the d α bound state is κ d α = √ µ d α ε d α = 0 .
31 fm − . The square of ANCfor the virtual decay Li → d + α is [ C ( d α )0 ] = 7 . − . This value is higher than the realistic value ofthis square of [ C ( d α )0 ] = 5 .
29 fm − [104]. To get thecorrect ANC from the one obtained in the Woods-Saxonpotential we need to introduce the spectroscopic factor.However, since we are not interested in the absolute crosssection, we keep using the ANC generated by the Woods-Saxon potential.Usually the indirect experiments are performed at fixedincident energy of the projectiles [3]. In the case un-der consideration the projectile is Li or C (in the in-verse kinematics). To cover the E α C energy interval5 FIG. 15. Square of the d − α bound state wave function inthe momentum space. First published in [60]. ∼ E Li C , oneneeds to change p d α . Since k Li is fixed to change p d α we have to change k d so that p d α ≤ κ d α . It can beachieved by changing k d or its direction ˆk d or both. Ex-perimentally one can select all the events falling into theregion p d α ≤ κ α d . Here, to simplify calculations, we as-sume that k d || k Li . It means that the variation of p d α is achieved by changing of k d . Owing to the energy con-servation by changing k d we can vary E α C but simul-taneously we change the d − α relative momentum p d α .The fully DCS given by Eq. (119) is proportional to the d − α bound-state wave function in the momentum space ϕ d α ( p d α ), which decreases with increase of p d α , see Fig.15.To avoid significant decrease of the fully DCS whencovering the E α C energy interval ≈ E Li C but not too close to the Coulombbarrier in the initial channel of the indirect reaction (16).Taking into account the fact that this Coulomb barrier is ≈ E Li C = 7 MeV. In this case for E α C = 2 . − resonance, p d α = 0 .
141 fm − while at E α C = 0 . p d α = 0 .
281 fm − . Hence, when covering the E α C energy interval from E α C = 2 .
28 MeV to the most ef-fective astrophysical energy for the process C( α, γ ) Othe square of the Fourier transform ϕ d α ( p d α ) drops bya factor of 2 .
97. Note that the drop of ϕ d α ( p d α ), whenmoving from E α C = 2 .
28 MeV to 0 . .
1. Notethat ϕ d α ( p d α ) appears because we consider the indirectthree-body reaction. There is another energy-dependentfactor ˜ W L , which is also result of the consideration of thethree-body indirect reaction. This factor will be consid-ered below.Our goal is to calculate the photon’s angular distri-butions at different E α C energies. It allows us to com- pare the indirect cross sections at higher energy E α C =2 .
28 MeV and the most effective astrophysical energy E α C = 0 . α − C ofthe binary sub-reaction (95), the indirect method allowsone to measure the fully DCS at E α C = 0 . . E α C = 0 . − subthreshold resonance and 1 − resonance at 2 .
423 MeV is constructive or distrac-tive because the pattern of this interference mayaffect the photon’s angular distribution.3. The third goal is to compare the relative contribu-tion of the E E B. Astrophysical factors for C( α, γ ) O First, to determine the parameters, which we use tocalculate the fully DCSs, we fit the experimental as-trophysical factors S E for the E S E for the E C( α, γ ) O reaction from[67]. We do not pursue a perfect fit and are mostly inter-ested in fitting energies below the 1 − resonance at 2 . E α C ≤ E − stateand the 1 − resonance, and one background state. Notethat in the R -matrix fits one must take into account allthe upper-lying 1 − levels which is practically impossible.Sometimes taking into account closest levels, subthresh-old and resonance ones in the case under consideration, isenough. But in all the previous publications when fittingthe E E + subthreshold resonance and 2 + resonance at 2 .
683 MeV.Our goal is to demonstrate the pattern of the fullyDCS using reasonable parameters. A more accurate fitcan be done when indirect data will be available. In ourfit, we kept fixed only the parameters of the subthresh-old resonances 1 − and 2 + while the parameters of thehigher lying resonances 1 − and 2 + were varying. Thefixed parameters are shown in Table II in parentheses.This table shows the set of the parameters used to fit theastrophysical factors S E and S E . These parametersare also used to calculate the fully DCS. E n is the energyof the n -th level.Note that in the R -matrix approach, which includesa few interfering levels, it is convenient to choose one6 TABLE II. Parameters used in calculations of the astrophysi-cal factors of the C( α, γ ) O radiative capture and the pho-ton’s angular distributions from the indirect C( Li , d γ ) Oreaction. L = 1 L = 2 E [MeV] ( − .
45) ( − . γ L L [MeV / ] (0 . . γ L ( γ ) 1 0 L L [MeV / fm L +1 / ] (0 . . E [MeV] 3 . . γ L L [MeV / ] 0 . . γ L ( γ ) 2 0 L L [MeV / fm L +1 / ] − . − . E [MeV] 33 . γ L L [MeV / ] 1 . γ L ( γ ) 3 0 L L [MeV / fm L +1 / ] − . of the energy levels to coincide with the location of theobservable physical state [105, 106] while energies of otherlevels become fitting parameters.We adopted E = − ε ( s ) α C(1) = − .
045 MeV for L = 1 and E = − ε ( s ) α C(2) = − .
245 MeV for L = 2transitions. Then the boundary condition for the sec-ond and third levels of the E E α C = − .
045 MeV while for L = 2 the boundarycondition is taken at E α C = − .
245 MeV. Moreover,because in our choice the locations of the subthresholdbound states for L = 1 and L = 2 are fixed, the energiesof other levels are fitting parameters and deviate fromthe real resonance energies. For example, the 1 − reso-nance at 2 .
423 MeV in the fit is shifted to E α C = 3 . + resonance at 2 .
683 MeV is shifted to 2 . − resonance at E α C = − .
045 MeV and the 1 − res-onance at E α C = 2 .
423 MeV does not contradict thefact that in the fit the resonance at 2 .
423 MeV is shiftedto 3 . E . − resonanceand resonance at 2 .
423 MeV at low energies. Changingthe sign of γ γ ) 2 0 1 L = − . / fm / to pos-itive provides the destructive interference between the first two 1 − levels. In what follows by the E − levels.In Fig. 16 the calculated S E and S E astrophysicalfactors for the E E S E (0 . . E S E (0 . . E E E S E (0 . C. Photon’s angular distributions
In Figs. 17 and 18 the photon’s angular distributionsare shown at four different E α C energies: 0 . , . , . .
28 MeV. We do not show the angular distributionsat an intermediate energy of 1 . E E Li C = 7 MeV (10 . Li as a projec-tile), which is higher than the Coulomb barrier V CB ≈ Li + C of the indirect reac-tion. The angular distributions are calculated in the c.m.of the reaction Li( C , d γ ) O. We selected axis z || p xA .Hence the photon scattering angle is counted from p xA ,which itself is determined by k s . The calculated photonangular distribution determines the angular correlationbetween the emitted photons from the intermediate ex-cited state F ∗ and the spectator s .Figures 17 and 18 are very instructive. First, we notethat the E ◦ while the E ◦ and 135 ◦ .However, the interference of the E E . E a ) and ( b ) in Fig. 17, with pronouncedpeaks at 52 ◦ and 50 ◦ , respectively. The character of thetotal angular distribution at 0 . E E . c ) and ( d ), are the most instructive. The patterns ofthe photon’s angular distributions are different for theconstructive and destructive E E E FIG. 16. Low-energy astrophysical S E ( E α C ) and S E ( E α C ) factors for E E C( α, γ ) Oradiative capture. Black dots are astrophysical factors from [67], solid red line is present paper fit. Panel (a): S E ( E α C )astrophysical factor; panel (b): S E ( E α C ) astrophysical factor. First published in [60].FIG. 17. (Color online) Angular distribution of the photons emitted from the reaction C( Li , d γ ) O proceeding through thewings of two subthreshold resonances 1 − , E α C = − .
045 MeV, 2 + , E α C = − .
245 MeV, and the resonances at E α C > E
1, the blue dashed line is theangular distribution generated by the electric quadrupole E E E E α C = 0 . E − , E α C = − .
045 MeV and the resonance 1 − , E R = 2 .
423 MeV; panel (b): E α C = 0 . E − , E α C = − .
045 MeV and theresonance 1 − , E R = 2 .
423 MeV; panel (c): the same as panel (a) but for E α C = 0 . E α C = 0 . FIG. 18. (Color online) Angular distribution of the photons emitted from the reaction C( Li , d γ ) O proceeding throughthe wings of the two subthreshold resonances, 1 − , E α C = − .
045 MeV, 2 + , E α C = − .
245 MeV, and the resonances at E α C >
0. Notaions of the lines are the same as in Fig. 17. Panel (a): the same as panel (a) in Fig. 17 but for E α C = 2 . E α C = 2 . E α C = 2 .
28 MeV; panel (d): the same as panel (d) in Fig. 17 but for E α C = 2 . interference is too small compared to the cross section at0 . E E E − levels. Hence, the angular distributions at higher energiescannot distinguish between constructive and destructive E − resonance at 2 .
28 MeV exceeds the one at 0 . .
28 MeV to 0 . . Our estimation shows that measurementsof the indirect fully DCS at 0 . C( α, γ ) O right at the most effective astrophysical en-ergy 0 . VIII. SUMMARY
In this review, we focused on the theory of the Trojanhorse method. The THM is a powerful indirect techniqueallowing to obtain the astrophysical factors for resonantrearrangement reactions at astrophysically-relevant en-ergies which often cannot be reached in direct experi-ments. The THM was suggested by Baur [4]. However,in its suggested form the method was not practical be-cause the low astrophysical energies of the binary subre-action x + A → b + B can be achieved at the high s − x relative p sx momentum. High p sx correspond to small r sx , meaning s is close to x and cannot be treated as aspectator. It was Spitaleri [14] who first described thecorrect kinematics making the THM method suitable forpractical applications. This kinematics follows from Eqs.(7)–(9). In particular, from Eq. (8) it follows that at thequasi-free kinematics ( p sx = 0), low E xA are achieveddue to the compensation by the binding energy of theTH particle a = ( sx ).Another important point regarding the application ofthe THM concerns types of the binary reactions whichcan be studied using the THM. There are two main typesof binary rearrangement subreactions playing an impor-tant role in nuclear astrophysics: direct and resonant re-actions. Due to the presence of the Coulomb-centrifugal9penetrability factors in each partial wave, the direct rear-rangement reactions at astrophysically-relevant energiesare contributed by the lowest partial waves l xA = 0 , x + A is absent. Hence, a significant numberof the partial waves can contribute to the THM ampli-tude of the subreaction x + A → b + B and, a priori,the THM cannot be used to extract S factors for the di-rect reactions except for some exceptional cases for which Q = Q = m x + m A − m b − m B is so large that condition k bB >> k xA takes place. Then the angular distributionof the DCS of the binary subreaction x + A → b + B willbe isotropic or almost isotropic, so that only l xA = 0 , x , which is transferred from the TH particle a to nucleus A to form a resonance state F ∗ , is virtual. Another diffi-culty is associated with the Coulomb interaction betweenthe particles, especially, taking into account that the goalof the THM is to study resonant rearrangement reactionsat very low energies important for nuclear astrophysics.The exact theory of such reactions with three chargedparticles is very complicated and is not available. Thatis why different approximations are being used to an-alyze THM reactions. The simplest one the plane-waveapproximation in which all the rescaterring effects are ne-glected. This approach has significant shortcomings andmay provide incorrect results, especially for heavier par-ticles. The shortcoming of the PWA has been discussedin this paper.In this review paper we described a new approachbased on a few-body approach, which provides a solid ba-sis for deriving the THM reaction amplitude taking intoaccount rescattering of the particles in the initial, inter-mediate and final states of the THM reaction. Since theTHM uses a two-step reaction in which the first step isthe transfer reaction populating a resonance state, we ad-dressed the theory of the transfer reactions. The theoryis based on the surface-integral method and R -matrix for-malism. The surface-integral approach allows one to takeinto account explicitly the off-shell character of the trans-ferred particle in the transfer reaction, which representsthe first step of the THM reaction. That is why the quasi-free character of the resonant sub-reaction is not needed.Moreover, the initial and final-state Coulomb-nuclear dis-tortions make the assumption of the quasi-free characterof the resonant sub-reaction invalid. To single out theamplitude of the resonance sub-reaction x + A → b + B ,which represents the second step of the THM reaction,we extrapolate the amplitude to the second energy sheetover E xA . We discussed application of the THM for a res-onant reaction populating both resonances located on thesecond energy sheet and subthreshold resonances, which are subthreshold bound states located at negative ener-gies close to thresholds. We also discussed the applica-tion of the THM to determine the astrophysical factorsof resonant radiative-capture reactions at energies so lowthat direct measurements can hardly be performed dueto the negligibly small penetrability factor in the entrychannel of the reaction. We elucidated the main ideas ofthe THM and outlined necessary conditions to performthe THM experiments. The shortcoming of the surface-integral based THM amplitude is that it neglects the con-tribution of the internal post-form DWBA part. Finally,it is assumed that the THM mechanism dominates, how-ever, no estimation of the background is given. It wouldbe interesting to investigate the accuracy of this assump-tion.Numerous examples are presented throughout the re-view to demonstrate practical applications of the THM.One of them is the analysis of the neutron generator reac-tion in AGB stars C( α, n ) O, which is contributed bythe subthreshold bound state. We also critically analyzedthe application of the THM method for the analysis of theone of the important astrophysical reactions, C + Cfusion. We showed that inclusion of the Coulomb-nucleardistortions in the initial and final states of the THM re-action used to determine the astrophysical S ∗ factor forthe carbon-carbon fusion drops S ∗ by three orders of themagnitude bringing its behavior with the agreement withthe theoretical prediction of the upper limit of the S ∗ factor. We considered also a new extension of the THM,namely, its application to radiative capture reactions. Itis demonstrated how this method can be applied to inves-tigate a crucially-important astrophysical radiative cap-ture reaction of C( α, γ ) O. We explained what mea-surements should be done to correctly identify the THMmechanism.
ACKNOWLEDGMENTS
A.M.M. acknowledges support from the U.S. DOEGrant No. DE-FG02-93ER40773 and NNSA Grant No.DENA0003841. A.S.K. acknowledges support from theAustralian Research Council and thanks the staff of theCyclotron Institute, Texas A&M University for hospital-ity during his visit. D.Y.P. acknowledges support fromthe NSFC Grant No. 11775013.0
Appendices
A. SPECTRAL DECOMPOSITION OF THETWO-CHANNEL GREEN’S FUNCTION
To single out the resonance in the subsystem F = x + A from the Green’s function G we follow Ref. [12]and rewrite it as G = G s (cid:0) V NxA G (cid:1) , (122)where V NxA = V Nsx + V NsA and G s ( z ) = 1 z − ˆ T sF − ˆ H F − V CxA . (123)Note that V CxA = V Csx + V CsA . Substituting Eq. (122) intoEq. (53) one gets M (cid:48) = (cid:10) Φ C ( − ) bB (cid:12)(cid:12)(cid:10) X f (cid:12)(cid:12) V bB G s ˜ U sA (cid:12)(cid:12) X i (cid:11)(cid:12)(cid:12) I ax Ψ C (+) k aA (cid:11) , (124)where the transition operator ˜ U sA is˜ U sA = V Nsx + V NxA
G V
Nsx . (125) Now we use approximation by replacing V CxA in Eq.(123) with U CsF to get G s ( z ) ≈ z − ˆ T sF − ˆ H F − U CsF , (126)where U CsF is the channel Coulomb potential describingthe interaction between the c.m. of nuclei s and F . Thenthe reaction amplitude M takes the form M (cid:48) = (cid:10) Φ C ( − ) bB (cid:12)(cid:12)(cid:10) X f (cid:12)(cid:12) V bB G s ˜ U sA (cid:12)(cid:12) X i (cid:11)(cid:12)(cid:12) I ax Ψ C (+) k aA (cid:11) . (127)Singling out the first step of the THM reaction a + A → s + F ∗ we get M (cid:48) = (cid:10) Φ C ( − ) bB (cid:12)(cid:12)(cid:10) X f (cid:12)(cid:12) V bB (cid:12)(cid:12) X f (cid:11)(cid:10) X f (cid:12)(cid:12) G s (cid:12)(cid:12) X i (cid:11) × (cid:10) X i (cid:12)(cid:12) ˜ U sA (cid:12)(cid:12) X i (cid:11)(cid:12)(cid:12) I ax Ψ C (+) k aA (cid:11) = (cid:10) Φ C ( − ) bB (cid:12)(cid:12) ˜ V bB (cid:12)(cid:12)(cid:10) X f (cid:12)(cid:12) G s (cid:12)(cid:12) X i (cid:11)(cid:12)(cid:12) U sA (cid:12)(cid:12) I ax Ψ C (+) k aA (cid:11) , (128)where the short-hand notations ˜ V bB = (cid:10) X f (cid:12)(cid:12) V bB (cid:12)(cid:12) X f (cid:11) and U sA = (cid:10) X i (cid:12)(cid:12) ˜ U sA (cid:12)(cid:12) X i (cid:11) are introduced. We assume thatthe potential V bB is spin-independent.To single out a resonance state in the intermediate sub-system F one can introduce the spectral decompositionof G s ( z ): (cid:10) X f (cid:12)(cid:12) G s (cid:12)(cid:12) X i (cid:11) = (cid:88) n (cid:90) d k sF (2 π ) (cid:12)(cid:12) I F n f Ψ ( − ) k sF (cid:11)(cid:10) Ψ ( − ) k sF I F n i (cid:12)(cid:12) E aA + Q n − k sF / (2 µ sF ) + i (cid:90) d k bB (2 π ) d k sF (2 π ) (cid:12)(cid:12) Ψ ( − ) k bB ; f Ψ C ( − ) k sF (cid:11)(cid:10) Ψ C ( − ) k sF Ψ ( − ) k bB ; i (cid:12)(cid:12) E aA − ε a + Q if − k bB / (2 µ bB ) − k sF / (2 µ sF ) + i . (129)Here we use the notion of the overlap function I F n f ( r bB ) = (cid:10) ϕ B ( ξ B ) ϕ b ( ξ b ) (cid:12)(cid:12) ϕ n ( ξ B , ξ b ; r bB ) (cid:11) , which is the projectionof the n -th bound-state of the many-body wave func-tion ϕ n of F on X f . Integration in the matrix ele-ment is carried out over all the internal coordinates ξ b and ξ B of nuclei b and B . Hence, I F n f depends only on r bB (for non-zero orbital angular momenta it depends on r bB ). Similar meaning has the second overlap function I F n i ( r xA ) = (cid:10) ϕ n ( ξ A , ξ x ; r xA ) (cid:12)(cid:12) ϕ x ( ξ x ) ϕ A ( ξ A ) (cid:11) introducedin Eq. (129).Usually the overlap functions are determined for boundstates. But here we also introduce the overlap functionsfor the continuum states. In particular, we defineΨ ( − ) k bB ; f ( r bB ) = (cid:10) ϕ B ( ξ B ) ϕ b ( ξ b ) (cid:12)(cid:12) Ψ ( − ) k bB ( ξ B , ξ b ; r bB ) (cid:11) (130)Ψ ( − ) ∗ k bB ; i ( r xA ) = (cid:10) Ψ ( − ) k bB ( ξ A , ξ x ; r xA ) (cid:12)(cid:12) ϕ A ( ξ A ) ϕ x ( ξ x ) (cid:11) (131)to be the projections of the wave function Ψ ( − ) k bB of the system F in the continuum on X f and X i , respectively.We assume that the continuum wave function Ψ ( − ) k bB hasthe incident wave in the channel f = b + B with k bB being the b + B relative momentum.Also in Eq. (129), E aA is the a − A relative kineticenergy, Q n = m a + m A − m s − m F n = ε F n − ε a , ε F n = m x + m A − m F n is the binding energy of the bound state F n for the virtual decay F n → x + A , Ψ C ( − ) k sF is theCoulomb scattering wave function of particles s and F with the relative momentum k sF , µ sF is the reducedmass of particles s and F , ε sx = m s + m x − m a isthe binding energy of a , m i is the mass of particle i , E aA − ε a + Q if is the total kinetic energy of the three-body system s + b + B , Q if = m x + m A − m b − m B , m i is the mass of particles i , i = x + A and f = b + B arethe initial and final channels of the binary subreaction x + A → b + B .Here, for simplicity, we consider the wave functionΨ ( − ) k bB only in the external region. The internal region can1be taken similarly using the R -matrix approach (see Ap-pendix A in [9]). In the external region the wave functionΨ ( − ) k bB with the incident wave in the channel f = b + B be-comes an external multichannel scattering wave function[9, 10]:Ψ ( − ) k bB ( r bB ) = − i k bB (cid:88) c (cid:114) v f v c r c X c × (cid:2) I ∗ ( k bB , r bB ) δ cf − S ∗ cf O ∗ ( k c , r c ) (cid:3) . (132)We recall that f stands for the channel b + B . The sumover c is taken over all open final channels c coupledwith the initial channel f . X c stands for the productof the bound-state wave functions of the fragments inthe channel c , v c is the relative velocity of the nucleiin the channel c , S c f is the scattering S matrix for thetransition f → c , O ( k c , r c ) is the Coulomb Jost singularsolution of the Schr¨odinger equation with the outgoing-wave asymptotic behavior.In the case under consideration we consider only twocoupled channels, i = x + A and f = b + B . In the ex- ternal region the channels are decoupled and the overlapfunction Ψ ( − ) k bB ; i is written asΨ ( − ) ∗ k bB ; i ( r xA ) = i k bB r xA (cid:115) µ xA k bB µ bB k xA S f i O ( k xA , r xA ) . (133)Equation (133) determines the projection of the externaltwo-channel wave function Ψ ( − ) F , which has an incidentwave in the channel f = b + B , onto the channel i = x + A .The second overlap function takes the formΨ ( − ) k bB ; f ( r bB ) = − i k bB r bB × (cid:2) I ∗ ( k bB , r bB ) − S ∗ f f O ∗ ( k bB , r bB ) (cid:3) , (134)where I ∗ ( k bB , r bB ) = O ( k bB , r bB ).We denote the second (continuum) term in Eq. (129)as (cid:10) X f (cid:12)(cid:12) G cont s (cid:12)(cid:12) X i (cid:11) : (cid:10) X f (cid:12)(cid:12) G cont s (cid:12)(cid:12) X i (cid:11) = (cid:90) d k bB (2 π ) d k sF (2 π ) (cid:12)(cid:12) Ψ ( − ) k bB ; f Ψ C ( − ) k sF (cid:11)(cid:10) Ψ C ( − ) k sF Ψ ( − ) k bB ; i (cid:12)(cid:12) E aA − ε a + Q if − k bB / (2 µ bB ) − k sF / (2 µ sF ) + i . (135)The resonant term corresponding to the subsystem F canbe singled out from Eq. (135). To this end1. We first perform the integration over the solid angleΩ k bB .2. The S -matrix element S i f has a resonance pole onthe second Riemann sheet at the b − B relative en-ergy E R( bB ) = E bB ) − i Γ /
2, where Γ is the totalresonance width. In the momentum plane this res-onance pole occurs at the b − B relative momentum k R( bB ) = k bB ) − i k I ( bB ) . We assume that the res- onance is narrow: Γ << E bB ) or k I (0) << k bB ) .3. When k bB → k R( bB ) the integration contour over k bB moves down to the fourth quadrant pinchingthe contour to the pole at k R( bB ) . Taking theresidue at the pole E bB = E R( bB ) one can singleout the contribution to (cid:10) X f (cid:12)(cid:12) G cont s ( z ) (cid:12)(cid:12) X i (cid:11) from theresonance term in the subsystem F .Following all these steps we get the desired spectraldecomposition of G s for two coupled channels [12]: (cid:10) X f (cid:12)(cid:12) G R s (cid:12)(cid:12) X i (cid:11) = − i π (cid:90) d k sF (2 π ) (cid:12)(cid:12) φ R( bB ) ( r bB ) Ψ C ( − ) k sF (cid:11)(cid:10) Ψ C ( − ) k sF ˜ φ R( xA ) ( r xA ) (cid:12)(cid:12) E aA − ε a + Q if − E R( bB ) − k sF / (2 µ sF ) , (136)where˜ φ R( xA ) ( r xA ) = e − i δ p ( k xA ) ) (cid:114) µ xA k R( xA ) Γ xA O ∗ ( k R( xA ) , r xA ) r xA ,φ R( bB ) ( r bB ) = e i δ p ( k bB ) ) (cid:114) µ bB k R( bB ) Γ bB O ( k R( bB ) , r bB ) r bB (137) are the Gamow resonant wave functions in channels i and f . Note that ˜ φ R( xA ) ( r xA ) is the Gamow wave functionfrom the dual basis, Γ xA and Γ bB are the partial reso-nance widths in the initial channel i and final channel f ,respectively, δ p ( k xA ) and δ p ( k bB ) ) are the potential(non-resonant) scattering phase shifts in the initial and2final channels.After deriving the expression for the resonance termin the spectral decomposition of (cid:10) X f (cid:12)(cid:12) G R s ( z ) (cid:12)(cid:12) X i (cid:11) we cansubstitute it into Eq. (128) and derive an equation forthe amplitude of the reaction a + A → s + b + B with threecharged particles in the final state, proceeding throughan intermediate resonance in the subsystem F = x + A = b + B . As mentioned above, the TH reaction amplitudeis described by the two-step process: the first step isthe transfer reaction populating the resonance state a + A → s + F ∗ and the second step is the decay of theresonance into two-fragment channel F ∗ → b + B leadingto the formation of the three-body final state, s + b + B . We derive below the expression for the TH reactionamplitude taking into account the Coulomb interactionsin the intermediate and final states.Substituting Eq. (136) into Eq. (128) and writing itin the momentum representation one gets M (cid:48) = − i µ sF π (cid:90) d p B (2 π ) d p b (2 π ) Φ (+)( bB ) k B k b (cid:0) p B , p b (cid:1) × W bB (cid:0) p bB (cid:1) J ( p sF , k aA ) , (138)where J ( p sF , k aA ) = (cid:90) d k sF (2 π ) Ψ C ( − ) k sF (cid:0) p sF (cid:1) M ( tr ) (cid:0) k sF , k aA (cid:1) k − k sF , (139)and M ( tr ) (cid:0) k sF , k aA (cid:1) = (cid:10) Ψ C ( − ) k sF ˜ φ R( xA ) (cid:12)(cid:12) U sA (cid:12)(cid:12) I ax Ψ C (+) k aA (cid:11) (140) is the amplitude of the transfer reaction a + A → s + F ∗ populating the resonance F ∗ . This amplitude can beapproximated by M ( prior ) M F M s ; M A M a ( k sF ) ˆk sF , k aA ) definedin Eq. (42). M ( tr ) = M ( prior ) M F M s ; M A M a ( k sF ) ˆk sF , k aA )The expression for the form factor W bB (cid:0) p bB (cid:1) can beobtained using Eq. (36) from [17]: W bB (cid:0) p bB ) = (cid:90) d r bB e − i p bB · r bB (cid:101) V bB ( r bB ) ϕ R( bB ) ( r pB )= e iδ p ( k bB ) ) e − πη R( bB )2 (cid:32) p bB − k bB ) k (cid:33) iη R ( bB ) × Γ(1 − iη R( bB ) ) (cid:115) µ bB Γ bB k R( bB ) g ( p bB ) , (141)where g ( p bB ) is the so-called reduced form factor, whichsatisfies g ( k bB ) ) = 1. Also k / (2 µ sF ) = E aA − ε a + Q if − E R( bB ) . (142)Note that the imaginary part Im( k R ) > B. DERIVATION OF THE THM RADIATIVECAPTURE AMPLITUDE
We analyze the THM radiative capture reaction (109)using the PWA. The derivation of the PWA for the trans-fer reaction is presented in [3] and in Section III E. For thecase under consideration, we use only the surface term,which takes the form: M M s M F ( s ) τ M a M A τ ( k sF τ , k aA ) = √ πµ xA i l xA ϕ a ( p sx ) (cid:112) µ xA R xA γ τ j xA l xA J F ( s ) ˜ W l xA (cid:88) M x m jxA m lxA (cid:10) J s M s J x M x (cid:12)(cid:12) J a M a (cid:11) × (cid:10) J x M x J A M A (cid:12)(cid:12) j xA m j xA (cid:11) (cid:10) j xA m j xA l xA m l xA (cid:12)(cid:12) J F ( s ) M F ( s ) τ (cid:11) Y ∗ l xA m lxA ( ˆp xA ) , (143)where ˜ W l xA is given by˜ W l xA = (cid:104) j l xA ( p xA r xA ) (cid:104) R ch ∂ ln[ O l xA ( k xA r xA )] ∂r xA − (cid:105) − R ch ∂ ln [ j l xA ( p xA r xA )] ∂r xA (cid:105)(cid:12)(cid:12)(cid:12) r xA = R ch + 2 µ xA Z x Z A α (cid:90) d r xA j l xA ( p xA r xA ) × O l xA ( k xA r xA ) O l xA ( k xA R ch ) . (144)This is the generalization of Eq. (41) by including the integral giving the external term contribution. Momenta p xA and p sx are defined by Eqs. (36).Note that M M s M F ( s ) τ M a M A τ ( k sF τ , k aA ) does not contain thehard-sphere scattering phase shift δ hsj xA l xA J F ( s ) . Also, ϕ a ( p sx ) is the Fourier transform of the radial part ofthe s -wave bound-state wave function ϕ a ( p sx ) of the a = ( s x ). Also, κ sx = √ µ sx ε sx is the wave number ofthe bound-state a = ( s x ), ε sx is its binding energy forthe virtual decay a → s + x . Since particles s and x arestructureless, the spectroscopic factor of the bound state a = ( s x ) is unity and we can use just the bound-state3wave function ϕ sx . Also k s and E xA are related by theenergy conservation [3]: E aA − ε sx = E xA + k s / (2 µ sF ) , (145)where k s is the momentum of the spectator s in the c.m.of the THM reaction, µ sF is the reduced mass of particles s and F .Now we consider the amplitude V ν , ν = 1 , , describ-ing the radiative decay of the intermediate resonance F ν → F + γ [105]: V M F M λM ( s ) Fν ν = − (cid:90) d r xA × (cid:10) I FxA ( r xA ) (cid:12)(cid:12) ˆJ ( r ) (cid:12)(cid:12) Υ ν ( r xA ) (cid:11) · A ∗ λ k γ ( r ) , (146)where I FxA ( r xA ) is the overlap function of the bound-state wave functions of x, A and the ground state of F = ( x A ). Again, for the point-like nuclei x and A theoverlap function I FxA ( r xA ) can be replaced by the single-particle bound-state wave function of ( xA ) in the groundstate. Also A ∗ λ k γ ( r ) is the electromagnetic vector poten-tial of the photon with helicity λ = ± k γ at coordinate r xA , ˆJ ( r ) is the charge current densityoperator. Matrix element in Eq. (146) is written assum-ing that on the first stage of the reaction the excited state F ν , ν = 1 , , is populated, which subsequently decays tothe ground state F .Using the multipole expansion of the vector potential,leaving only the electric components with the lowest al-lowed multipolarities L and using the long wavelengthapproximation for ˆJ ( r ) we get (see Ref. [105] for details) V M F M λM F ( s ) ν ν = √ π (cid:88) L i − L ( − L +1 (cid:112) ˆ L k L − / γ [ γ J LF ( s ) ( γ ) ν J F L ] × (cid:2) D LM λ ( φ, θ, (cid:3) ∗ (cid:10) J F M F L M (cid:12)(cid:12) J LF ( s ) M LF ( s ) ν (cid:11) , (147)where γ J LF ( s ) ( γ ) ν J F L is the formal R -matrix radiative widthamplitude for the electric ( EL ) transition J LF ( s ) → J F given by the sum of the internal and external radiative width amplitudes, see Eqs (32) and (33) from [41], inwhich we singled out √ k L +1 / γ . Since now we take intoaccount a few multipolarities L , we replace the previouslyintroduced spin of the intermediate resonance J F ( s ) by J LF ( s ) , where the superscript L denotes the multipolarityof the EL transition to the ground state F . Replacementof J F ( s ) by J LF ( s ) takes into account that the spins of theintermediate excited states are different for different mul-tipolarities. Since we added the superscript L to the spinof the intermediate resonance we added the same super-script to its projection M LF ( s ) ν . Also in Eq. (147) M isthe projection of the angular momentum L of the emittedphoton (multipolarity of the electromagnetic transition).We remind that V M F M λM F ( s ) ν ν does not depend on the hard-sphere scattering phase shift.The determined radiative width amplitude is relatedto the formal resonance radiative width by the standardequation Γ J LF ( s ) ( γ ) ν J F L = 2 k L +1 / γ ( γ J LF ( s ) ( γ ) ν J F L ) . (148)Note that the observable radiative width is related to theformal one by [see also Eq. (93)] (cid:18) ˜ γ J LF ( s ) ( γ ) ν J F L (cid:19) = ( γ J LF ( s ) ( γ ) ν J F L ) γ ν j xA l xA J F ( s ) (cid:104) d S lxA ( E xA )d E xA (cid:105) E xA = E ν . (149)We consider the two-level approach with ν = 1 ( ν = 2)corresponding to the subthreshold resonance (the reso-nance at E xA > E ν = − ε ( s ) xA for ν = 1 and E ν = E R for ν = 2 with E R being the resonance energycorresponding to the level ν = 2. This observable radia-tive width is related to the observable resonance radiativewidth as ˜Γ J LF ( s ) ( γ ) ν J F L = 2 k L +1 / γ (˜ γ J LF ( s ) ( γ ) ν J F L ) . (150)Substituting Eqs. (143) and (147) into Eq. (110) weget the expression for the indirect reaction amplitude M M s M F M λM a M A = ϕ a ( p sx )2 (cid:115) R xA π µ xA (cid:88) L ( − L +1 ˆ L / k L − / γ (cid:2) D LM λ ( φ, θ, (cid:3) ∗ (cid:88) l xA i l xA − L ˜ W l xA × (cid:88) ν, τ =1 γ J LF ( s ) ( γ ) ν J F L A Lν τ γ τ j xA l xA J LF ( s ) (cid:88) M LF ( s ) ν (cid:10) J F M F L M (cid:12)(cid:12) J LF ( s ) M LF ( s ) ν (cid:11) × (cid:88) m jxA m lxA M x (cid:10) j xA m j xA l xA m l xA (cid:12)(cid:12) J LF ( s ) M LF ( s ) τ (cid:11) (cid:10) J x M x J s M s (cid:12)(cid:12) J a M a (cid:11) × (cid:10) J x M x J A M A (cid:12)(cid:12) j xA m j xA (cid:11) Y ∗ l xA m lxA ( ˆp xA ) . (151)4The amplitude M M s M F M λM a M A describes the indirect re-action proceeding through the intermediate resonances,which decay to the ground state F = ( x A ) by emittingphotons. Equation (151) is generalization of Eq. (110)by including the sum over multipolarities L correspond-ing to the radiative electric transitions from the interme-diate resonances with the spins J LF ( s ) to the ground state F with the spin J F . Note also that we assume that twolevels contribute to each transition of multipole L . It re-quires the two-level generalized R -matrix approach. Thegeneralization of Eq. (151) for three- or more-level casesis straightforward. In Eq. (151) the reaction part andradiative parts are interconnected by the R -matrix levelmatrix elements A Lν τ .The sum over ν and τ in Eq. (151) is the standard R -matrix term for the binary resonant radiative-capturereaction. However, we analyze the three-body reaction a ( x s ) + A → s + F + γ with the spectator s in the fi-nal state rather than the standard two-body radiative-capture reaction x + A → F + γ . This difference leads tothe generalization of the standard R -matrix approach forthe three-body reactions resulting in the appearance ofthe additional factor ϕ a ( p sx ) W l xA which should be famil-iar to the reader from Eq. (74). That is why we call thedeveloped approach the generalized R -matrix method forthe indirect resonant radiative-capture reactions.We take the indirect reaction amplitude at fixed pro-jections of the spins of the initial and final particles in-cluding the fixed projection M of the orbital momentum L of the emitted photon and fixed its chirality λ . For ex-ample, for the C( α, γ ) O reaction the electric dipole E L = 1) and quadrupole E L = 2) transitions docontribute and they interfere. In the long-wavelength ap-proximation only minimal allowed l xA for given L doescontribute. For example, for the case considered below l f = 0 l xA = L = 1 for the dipole and l xA = L = 2 forthe quadrupole electric transitions do contribute. Thedimension of the R -matrix level matrix A L depends onthe number of the levels taken into account for each L .The indirect reaction amplitude depends on the off-shell momenta p sx and p xA . Both off-shell momenta areexpressed in terms of k a and k s , see Eq. (36). Also thethe indirect reaction amplitude depends on the momen-tum of the emitted photon k γ whose direction is deter-mined by the angles in the Wigner D -function.In the center-of-mass of the reaction (16) neglecting therecoil effect of the nucleus F during the photon emission from the energy conservation we get E aA + Q = E sF + k γ , (152) k γ = E xA + ε xA , (153)where E sF = k s / (2 µ sF ) , Q = ε xA − ε sx and ε xA is thebinding energy of the ground state of the nucleus F .To estimate the recoil effect we take into account thatin the center-of-mass of the reaction (16) the momentumconservation in the final state gives k (cid:48) F = − k γ − k s , (154)where k (cid:48) F is the momentum of the final nucleus F afteremitting the photon. Then the energy conservation leadsto E aA − ε sx = k s µ sF + E xA = k s m s + ( k (cid:48) F ) m F + k γ (155)= k s µ sF + 2 k s k γ m F cos θ (cid:48) + k γ m F + k γ . (156)We remind that we use the system of units in which (cid:126) = c = 1, hence, E γ = k γ . Clearly, the term k γ m F = E γ E γ m F can be neglected because E γ << m F . The contributionof the term 2 k s k γ m F cos θ (cid:48) depends on cos θ (cid:48) = ˆk s · ˆk γ .Neglecting the recoil effect of the nucleus F in Eq.(155) we can replace k (cid:48) F by k s . Then k γ and k s are relatedby Eq. (152) while k γ and E xA are related by Eq. (153).If we would take into account the recoil effect then therelationship between k γ and E xA is more complicatedthan Eq. (153) and is given by k γ = E xAk s cos θ (cid:48) m F + 1 , (157)where we neglected the extremely small term k γ / (2 m F ).The expression for p xA is needed to calculate ˜ M l xA .From the energy-momentum conservation law in thethree-ray vertices a → s + x and x + A → F ( s ) of thediagram in Fig. 14 we get [3] E xA = p xA µ xA − p sx µ sx − ε s x . (158)In the QF kinematics p sx = 0 and E xA = p xA µ xA − ε s x . 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